Now that it has been reported that all charges against jussie Smollett will be dropped, does the federal government have the right to charge him on their own? Also, if it is found that there was impropriety in the state prosecutors office, will the charges automatically be restored?
Federal crimes and State crimes are different crimes - double jeopardy doesn’t apply, you can be charged with both.
Prosecutors have wide discretion on who they charge and who they don’t. If it is demonstrated that a prosecutor has acted corruptly then that case will be referred to a different prosecutor - who has wide discretion etc.
The United States has dual sovereignty which means that if it's a crime in two different jurisdictions both jurisdictions can try the case. So the crime was committed in Chicago, and goes to local and state agencies in Cook County, Illinois. But it's also a crime to falsify a criminal report in the U.S. Federal Government, so they can take a swing at it too.
As a general rule, the DOJ does not prosecute crimes for which the state has prosecuted (regardless of outcome at trial... if you're declared innocent in Illinois, the Feds rarely will try you again... there's actually a set of specific conditions, called the Petite Policy, they have that governs whether they will prosecute you, though these are self imposed by the DOJ, not a law governing them) but they do have the option, and often use it if there seems to be any funny business on the state's part to give a more favorable outcome to the defendant than he or she should have reasonably had.
Additionally, one of the crimes alleged against Smollett is that he mailed himself a threatening letter (containing a white powder). As any crime involving the mail is instantly elevates it to a Federal Crime, Smollett has not escaped possible charges from this investigation as of time of writing. Federal Investigations tend to be much more thorough than state level investigations so it's possible that this aspect of the crime could result in new charges that Chicago wouldn't have jurisdiction over. The Petite Policy does not limit the Federal Government when the charges are not brought by the state, so even if a trial had occurred, the Feds could charge him with crimes related to the letters and not violate Double Jeopardy or Petite Policy.
Finally, while it is usually best understood that Double Jeopardy attaches when the jury is seated, this is not necessarily true. Although rare, if it was found that the trial was conducted corruptly in the defenses favor (including Jury Tampering) Double-Jeopardy is not attached, as the defense was never in a state of Jeopardy to begin with. If this is proven, a second trial on the same crimes can be held with a fresh panel of Jurists. It's rare, and certainly did not occur here (because no trial) but it's happened before.
As to the nature of the possible corruption on the prosecution's part, the charges will not automatically reinstate, as the acting-prosecutor will still have proprietorial discretion and will need to decide if he/she thinks there is a strong case. In all likelihood, the Feds may charge Smollett with the crimes covered under the Petite Policy, if only to remove trial from the local area to avoid the contentious nature of the facts interfering with justice. This is more of my own guess than actual fact, so take with salt, but it's probably the likely that there will be questions as to whether the prosecutor's office can still handle the trial.