i would like to know: if there is difference between the type of cases
for which something can be settled between the victim, and the
suspect, by money, or just by forgiveness . And cases for which one
will remain persecuted by the state, even-though the victim or its
family did not file a charge, or request to dismiss a charge?
In nearly universal United States criminal law jurisprudence, the victim of a crime does not have any authority to settle a criminal case for any type of crime for any reason. The distinction you are looking for pretty much does not exist in U.S. criminal law.
All criminal prosecutions in the United States are brought by "the People" or "the State" or in the name of "the People" or "the State" even in the small minority of states (not including California) in which private individuals can bring and prosecute criminal complaints (Massachusetts is one of the outliers in the U.S.; England also allows private individuals to prosecute some crimes in a similar fashion, although the government can take over a case if it wishes to do so.) In countries in the British Commonwealth that still recognize Queen Elizabeth as their symbolic head of state, criminal prosecutions are brought in the name of "the Crown."
Most crimes simultaneously give rise to civil liability for money damages, and to criminal liability. A settlement with the victim resolves civil liability for money damages, but not criminal liability.
A victim can, in practice, not complain about a crime to law enforcement, and if no one ever tells law enforcement, the criminal justice system will never learn about it prior to the death of the perpetrator or the expiration of the statute of limitations, and the crime will never be prosecuted.
A victim can request that the prosecutor assigned to a case drop charges or be lenient, and often, a prosecutor's office will defer to a victim's wishes, even though it is not required to do so. This is true even in the many states where victim's right statutes give victims a right to share their opinions about how a criminal case should be handled with prosecutors and judges.
A victim can refuse to cooperate with law enforcement in prosecuting a case, although there can be negative consequences to the victim if the victim does so. But, often when a victim doesn't cooperate, the criminal justice system gives up and drops a case because it isn't worth their trouble when other cases have cooperative victims and criminal justice resources are scarce.
Indeed, all these kinds of actions are often taken by victims in domestic violence cases.
But, these requests by a victim are not legally enforceable and an agreement by a victim to do or not do these things is not legally enforceable. It is unethical for a U.S. lawyer to offer to settle a case in a manner that involves this kind of conduct by a victim, and in rare cases a lawyer could be disbarred for doing so. In rare cases (which still, really do happen sometimes, often, for example, in statutory rape and domestic violence cases) a victim can be held in contempt of court and incarcerated for not being willing to testify in court against a suspect. Victims of crimes have no legal right to remain silent in U.S. law, unless they are also at risk of criminal liability based upon their testimony.
A victim can also lie to law enforcement or a court about what happened. In Les Miserables by Victor Hugo, the clergyman lies and says that property that was stolen by our protagonist was actually given as a gift so that he would not be punished criminally. Similarly, a rape victim can tell a law enforcement officer or a court that sex was consensual, even when it was not, in order to protect the victim. But, of course, lying to law enforcement officers in an official criminal investigation, or lying to a court during a court proceeding, is a crime, even by a victim of that crime, even though lies of that type are rarely prosecuted. (In Colorado, a state with five million people, there are only a dozen or so perjury prosecutions per year in the entire state even though people lie in court and on documents subject to felony perjury prosecution if someone lies on them hundreds of times a day, if not more often.)
But, in U.S. criminal law, even when a victim's lack of consent is an element of the crime, such as for the crimes of theft or rape, criminal liability can not be eliminated by having the victim ratify the perpetrator's conduct after the fact.
There are countries in Europe, for example, Germany, which do have some categories of crimes that a victim can settle unilaterally without the consent of a prosecutor. Such crimes are commonplace in Islamic law even for serious offenses. Pre-Roman Germanic customary law, e.g. in Medieval Iceland, recognized this concept which generally fell under the rubric of blood money which is quite similar to modern Islamic law on the topic.
Many countries in the civil law tradition historically had or still have a law that allows a rape victim to erase the criminal rape charges by marrying the victim. But, I won't speak to those differences in those systems between crimes that can be settled by victims and crimes that cannot be settled by victims, as this is not my areas of expertise.
Similarly, in many European countries, for example, Germany, a victim has a right to demand that a crime be prosecuted, while it is extremely rare in the U.S. for victims to have such a right, although some states allow a victim to demand that a special prosecutor be appointed to investigate whether a criminal case should be brought when the usual prosecutor declines to do so, and a small minority of states allow victims to personally prosecute certain minor crimes, as noted above.
The authority of a prosecutor to prosecute a crime over the objections of the victim, or to decline to prosecute a crime despite a victim's insistence that it be prosecuted (even when the victim can prove that the crime was committed by someone) is called prosecutorial discretion and is very nearly absolute in U.S. criminal law jurisprudence.
If you is an uninvolved witness of one of these a crime examples. For
which of these cases are you obliged to report your observations to
the police by law?
If you are not a witness but you "heared" from person B about person A
doing a thing mentioned in the previous cases, while person B did not
report this to the police, are you obliged to report this?
For what it is worth, this is not very closely related to the first question under U.S. law.
This would vary between U.S. states and between foreign jurisdictions.
The default rule in U.S. law is that there is no duty to report a crime unless you are served with a subpoena and required to testify. If you are served with a subpoena you are legally required to testify and can be fined or incarcerated if you do not, unless an evidentiary privilege (e.g. attorney-client privilege, clergy-confessional privilege, doctor-patient privilege, spousal privilege) applies and most of those evidentiary privileges have exceptions in some circumstances.
Most U.S. states do have statutes that require certain kinds of professionals who routinely deal with children (e.g. teachers, pediatricians, nurses, child care center workers) to report suspected child abuse.
A minority of U.S. states (e.g. Texas) have statutes that require members of the general public to report felonies.
In states that have these laws either in general or involving a limited group of people with special reporting responsibilities, there is typically a rather elaborate jurisprudence concerning how much knowledge triggers a duty to report. For example, a citizen may see something suspicious but not know for sure that it involves a crime, or at least, that the crime involved is actually a felony, rather than a misdemeanor.
The instance you cite of someone who is aware of hearsay evidence that a crime was committed, but doesn't have personal knowledge that a crime was committed, presents another distinction that might be made.
Similar issues come up in the case of lawyers who usually have a legal duty to report violations of the rules of professional conduct of which they are aware.
Another question that comes up in places that have these requirements is how quickly the crime that is discovered must be reported.
But, even when there is a legal duty to report a felony, it is rarely enforced and none of the jurisdictions where I have practiced law have such a statute, so I am not familiar with those distinctions in any particular jurisdiction in a criminal law context.
