As I understand - IANAL:
If you are charged, and then go to court, you have the choice to testify or not. If you choose to testify, then you have “opened the door”. You cannot say “I didn’t do it” and then refuse to answer on cross examination any details which may prove you wrong.
So the scenario where you are in court and plead the fifth would be at someone else’s trial, where you may provide testimony that could lead to charges against you. You can also plead the fifth in police interrogations and depositions leading up to the trial (as mentioned above, you must explicitly say that you are pleading the fifth.) The fact that you did so cannot be mentioned in court, since the implication is “this must mean he is guilty…” while again, as mentioned above, the facts you are hiding simply may imply that, even though you are innocent.
Quite often the prosecution will give A an immunity deal to get damning testimony about B. Then A cannot plead the fifth. What the immunity covers is part of the deal.
As I understand, if the other side objects, the judge can order an in camera session to have the details of what is involved explained without it going on the record, and then rule whether it lets the witness plead the fifth.
You might also be involved in a civil case. You are not obliged to answer, may plead the fifth, if the answers you give in civil trial may provide details that could lead to charges. (Again, even if you are innocent). However, the problem as I understand is this - if you fail to answer in civil court, this can be held against you; this is not a determination of guilt, it is a trial to see who is at fault. If you cannot provide enough information to show that you are the one most likely in the right - preponderance of evidence - then you lost the case. You are being sued for driving drunk and causing an accident.
“Did you have more than two drinks at the Happy Time bar that night?”
“I plead the fifth.”
If the plaintiff can show by other means that you likely did (i.e. bar tab) and you cannot refute that, then you lose.
Not so much the plead, but rather the lack of positive evidence you could have given.
And if your criminal trial is already settled, then what you say in a civil case cannot affect the result due to double jeopardy. If you are already found not guilty of DUI (or guilty), you have no further criminal legal jeopardy for testifying you were drunk, so no grounds to plead the fifth. Hence, waiting until after the criminal trial means less doubt from the defendant’s testimony in the civil trial - he would have to answer. (Of course, if a guilty verdict is being appealed…)
What’s not clear in this, is when there is both federal and state violations. Presumably you would need immunity from both to be free from prosecution for your testimony.
And I assume a Congressional Inquiry can play video of a witness taking the fifth and let people assume what they will, since again this is not a criminal trial in front of a judge or jury.