Short Answer
This account would be a fairly extreme outlier relative to normal practice in jury selection, but it is certainly something that could possibly happen (except for one small detail that isn't very relevant to the core issues that it raises; this detail is discussed below in the last heading of this answer).
The substantive points made in the Facebook post about what the facts recounted say about the state of women's conditions and attitudes about women in our society, and the poster's skepticism that a judge who made decisions like these really acted appropriately, are well founded, fair, and appropriate to raise. These concerns are within the heartland of what the case law on these kinds of issues discusses and struggles with, although because of the procedural posture of this issue, there isn't a lot of case law on this issue.
Long Answer
My Sources Of Knowledge
I have participated in choosing perhaps half a dozen juries in which I was counsel, and I have observed the process in other cases one or twice and have previously been in a couple of jury pools myself (I've never been actually selected to serve).
I'm also familiar with accounts of others practitioners regarding their experiences with jury selection, I know what I was taught in law school about the process, and I am familiar with the academic and practitioner oriented literature about the process.
I am answering based upon U.S. law, because I don't have intimate familiarity with petite jury selection in other common law countries, although the broad outlines of the process are similar.
As a caveat to this answer, however, recognize that judges have very broad discretion in the jury selection process and that not all judges adhere to "best practices". Further, in many states, judges are partisan elected officials who tend to end up in there positions because of, rather than in spite of, their extreme views and positions of legal issues over which they have discretion relative to the average lawyer or judges in places where the judicial selection process is less partisan.
Your Questions
How often are jury members subject to separate examination by defense
counsel (or prosecution counsel) during jury selection?
This usually happens in any case more serious than a simple traffic offense. It would be unusual to examine every single member separately in a separate room, but normally quite a few jurors a questioned separately on one point or another. This process is called voir dire (a situation in which U.S. legal terminology borrows from French rather than Latin).
How frequently are jury members dismissed in the presence of other
members?
Jury members are usually dismissed in the presence of other members, but the specific juror responses to the reasons for doing so may or may not be discussed in the presence of the other jurors. Less sensitive questions (e.g., do you still live in this county, are you a U.S. citizen, do you speak English, do you have scheduled health procedures during the anticipated trial or a trip with a non-refundable ticket or similar issue, and a few other "categorical" exclusions) are usually discussed in the presence of other members, as are responses to general questions that are unlikely to be grounds by themselves for a dismissal for cause, but could inform peremptory challeges of jurors.
The better and more common practice is for sensitive questions, such as a previous incident of being assaulted, to be discussed out of earshot from the other panel members, but in the hearing of the judge, at least one lawyer for each side and the court reporter who includes the bench conference in the trial transcript.
Alternatively, how often are dismissed jury members informed of the
selection or elimination of fellow jury members?
Normally, jury pool members leave the courtroom and get on with their day once they are dismissed. But, if they choose to stick around, they will learn who is selected to serve on the jury, and who is eliminated from the jury pool.
Is the defense counsel able to eliminate (all twelve) jury members
from consideration (for cause) without objections by prosecution
counsel?
Terminology
First of all, here and in your prior questions it is important to distinguish between the jury pool (a group of randomly chosen people who might end up being jurors) and the jury (a group of people ultimately selected to decide a case after challenges for cause and peremptory challenges are complete).
The Voir Dire Process
There is, in principle, no limit on the number of members of the jury pool who can be dismissed for cause, and if they run out, everyone goes home for the day and comes back the next day with a supplemental batch of jury pool members. In an obscure civil case or minor low profile felony case, the jury pool would typically be 30-60 potential jurors. In a death penalty case or case that has famous parties (e.g. a civil case involving Taylor Swift held in Denver recently), the jury pool would typically be several hundred people and the first round of voir dire would happen over the course of several days or even weeks.
Any time that either party requests that a juror be dismissed for cause, the other party can choose to not object, or can object and argue that the juror should not be dismissed for cause.
The judge doesn't have to dismiss a juror even when both parties agree to strike the juror for cause and not infrequently will refuse to dismiss a juror even when both parties agree to strike a juror for cause. This is because the judge has an institutional incentive to discourage jurors other than the one that a party has moved to dismiss from the jury pool from giving a lame excuse to try to get out of jury duty, even if the parties, who don't have that institutional concern, don't care about that.
Why Might A Prosecutor Fail To Object?
The downside of objecting to a request to dismiss a juror for cause is that it creates an almost automatic appellate issue for the party seeking to dismiss the juror if the juror is not dismissed.
So, a prosecutor might not object to a questionable request to dismiss a juror for cause in order to reduce the likelihood that a conviction obtained by the prosecutor would be reversed on appeal. Giving the defense the jury it wants also makes it more likely that if the case starts going badly that the defendant will agree to a plea bargain mid-trial rather than risking a conviction by the jury, because any conviction obtained is more likely to hold up on appeal.
A defendant may appeal a ruling denying a motion to dismiss a prospective juror that is denied after a conviction, if any, is entered.
If the jury acquits the defendant (the unfavorable outcome the prosecutor would like to avoid by not having a juror dismissed for cause), the prosecutor can't appeal the case, and if the jury hears evidence, the case can't be dismissed without prejudice or retried unless there is a conviction that is reversed or there is a mistrial (the mistrial rules are bit complicated).
So, if the prosecutor was really appalled by the dismissal for cause of so many women and felt that this would impair the prosecution's chance of obtaining a conviction materially, the prosecutor would have to dismiss the criminal charges so as to vacate the trial, before evidence was presented to the jury, and then refile the charges (assuming that this would be possible consistent with statutes of limitation and speedy trial requirements).
But, this would be an extraordinary move with high stakes, because the prosecutor has a long term strategic interest in not pissing off a judge in any case because that could cause the judge to exercise the judge's discretion against the prosecutor in future cases. The judge and prosecutor may have to deal with each other in future cases for decades and will do so on a regular basis every few weeks or months.
A judge is likely to be pissed off in this situation because dismissing a case ready to go to trial and scheduled for trial with a jury fully selected because the prosecutor was unhappy with the judge's rulings on motions to dismiss jurors for cause would not be appreciated by the judge who naturally believes that the rulings made on those motions were sound even if that belief is unreasonable.
I strongly suspect that this was the reason that the prosecutor allowed all twelve women in the jury panel to be stricken for cause in the case that you describe (assuming, of course, that the Facebook account is factually accurate, which is sometimes the case and sometimes not the case – even if the gist of the account was accurate, it wouldn't be surprising if some technical details or nuances were incorrectly recounted).
It Would Not Be Normal For A Prosecutor To Not Object In This Case
Despite these procedural considerations, it would be very unusual for a prosecutor to not object to striking all twelve women on a jury panel for cause in these circumstances and it would be very unusual for a judge to agree to strike all twelve women on the jury panel for cause in these circumstances whether or not the prosecutor objected.
Generally speaking, merely having had a prior experience of having been assaulted would not be sufficient to strike a prospective juror for cause. Normally, the prosecutor and/or the judge would ask the prospective juror if this experience made it impossible for that particular juror to be impartial in this particular case, and normally most of the prospective jurors asked that question would say "no". Usually, in that situation, the judge would not agree to dismiss that prospective juror for cause.
Most prosecutors would expect their objections to a request to dismiss a prospective juror for cause in this situation to be taken seriously by the judge and for only a few of these requests that cast the most doubt on the impartiality of a potential juror to be granted. And, most prosecutors would not consider the appellate risk involved in opposing a request to dismiss a prospective juror for cause in the typical scenario that I outlined above very troubling, because a judge has fairly broad discretion on dismissals of jurors for cause in the face of a marginal fact pattern.
The fact that the judge allowed this also suggests that the prosecutor may know that the judge is very unenlightened and has misogynist leanings and that fighting the judge's ruling in this case would be a lost cause that is hard to appeal.
(Of course, if any of the women had previously been assaulted by the defendant in this particular case and personally knew that defendant well, that would normally cause the potential juror to be dismissed for cause.)
After Challenges For Cause
Keep in mind also that after dismissals for cause are completed, a certain number of jurors equal to the number of peremptory challenges allowed to the prosecution and defense combined plus the number of jurors who need to be left over to decide the case would be put in the second stage of narrower jury pool. In this second stage, during which the narrower jury pool is honed to the actual final panel of jurors who will hear the case, each side exercises their allocated number of peremptory challenges (normally alternating back one forth, one juror at a time).
While peremptory challenges can generally be made without good cause, you can not make a peremptory challenge solely based upon a potential juror's race or sex. If a side dismisses all women, or all men from the jury pool, there is a presumption that this is what was done by the party striking the jurors that must be overcome with convincing reasons not based on race or sex.
The same analysis, strictly speaking, doesn't apply to motions to dismiss jurors for cause because in those cases a non-discriminatory reason has been definition been advanced by the party seeking to dismiss the prospective juror and accepted by the judge as convincing based upon the voir dire evidence.
One reason not to fight very hard to dismiss a juror for cause is that many of those jurors who seem most favorably inclined to your case, although not necessarily all of them, are likely to end up being dismissed in a peremptory challenge in any case.
A Footnote on Ex Parte Proceedings
Ex Parte Voir Dire Is Improper
The author of the OP also clarified that:
When I asked, "How often are jury members subject to separate
examination […]?", I meant that only one party (or counsel for one
party) is examining the jury at a time. E.g., first the judge and
prosecution counsel examines the jury (without defense present) then
the judge and defense counsel examines the jury at some other time.
Normally, the judge asks boilerplate routine questions first, then one side asks questions, then the other side asks questions.
But, it would be almost unheard of (and it would be improper and unethical) for this to happen without lawyers for both sides present to see what transpires during the other side's questioning, even if the prosecution wasn't planning to make challenges for cause, so as to gather up information needed for the preemptory challenge phase of jury selection.
To have a proceeding without both sides having a lawyer present is called an ex parte communication with the judge which both the judge and the lawyer doing so have an "ethical" duty to avoid in this part of the jury selection process.
An ethical duty means that the judge could be, in theory, kicked of the bench or temporary suspended or public reprimanded or require to take a judicial ethics class for doing so; and that the lawyer could similarly be disbarred or suspended from the practice of law or publicly reprimanded or required to take a legal ethics class for doing so.
Conducting the proceeding ex parte could also be grounds for a mistrial ruling that would not prevent the defendant from being retried (because it would have been done at the request of and for the benefit of the defense counsel), or for an appeal in the appropriate cases.
(There are ex parte proceedings which are ethical and permitted such as applications for search warrants and arrest warrants, but juror selection is not a proceeding to which an exception to the general rule applies.)
The Prosecution Could, In Theory, Waive The Right To Participate
In principle, the prosecutor could waive the right to be present while the judge and defense lawyer examine prospective jurors, but this would be almost bizarre conduct that would only happen if someone was calling about an incredibly urgent development that threatened to shut down the courts or put the prosecutor or co-workers or family at immediate risk of physical harm happened.
For example, a prosecutor might waive a right to be present if the prosecutor wasn't planning on objecting to any jurors for cause and was learning that the 9-11 attacks were underway, or that a mass shooting at the prosecutor's offices had happened or was in progress, or a prosecutor having a spouse who the person they stepped out to talk with was in a newly discovered hostage situation, or the prosecutor suddenly feeling an intense need to vomit or being on the verge of passing out or feeling like a heart attack might be in progress.
Even then, however, it would be more common for a judge to call a recess for a little while in any of those circumstances.
The Case Description Is Probably Incorrect Or Misleading On This Point
The ex parte examination of prospective jurors, which was apparently described, would be so far outside the norm of typical jury selection conduct, that I suspect that this part of the account is inaccurate or was confusingly worded.
For example, the post's wording arguably misleadingly implied that there was an ex parte proceeding as part of the jury selection process, but it wouldn't be necessarily inconsistent with a scenario in which a prosecutor was already present with the female jurors "before the remaining candidates were led into a room to meet the judge and defense attorney."
For example, it could be that what really happened was that a member of the prosecutors office was introduced to the in the jury assembly room, and led the prospective jurors to the correct courtroom without making any comments to them, and then they met the judge and defense counsel, which would be entirely proper and not unusual.
