^^^^

Apparently you think law suits are not legal investigations that can lead to criminal charges? Ask Bill Clinton. Of course Trump's enemies are fishing. They will for his entire administration, no matter how short that may be.
There is no law in any jurisdiction that would require Trump to answer questions during discovery in a civil action when his response might be self-incriminating. He would simply assert his Fifth Amendment rights. Here is a good explanation of a defendant's rights to invoke the Fifth Amendment in criminal and civil proceedings.
“What happens if you invoke the privilege against self-incrimination in a civil case?
“1. You can do it, and you won’t be held in contempt for failing to testify. Though the provision says that no person “shall be compelled in any criminal case to be a witness against himself,” the Supreme Court has made clear that this extends to compelling a person to testify in a civil case, when that compelled testimony could later be used against him in a criminal case. See, e.g.,
McCarthy v. Arndstein (1924):
“2. But a decision to take the Fifth may be used against a party in a civil case (if the party is the witness who refuses to testify, or is closely enough connected to the witness). In a criminal case, the judge and the prosecutor may not tell a jury “that it may draw an inference of guilt from a defendant’s failure to testify about facts relevant to his case.” But that’s not so in a civil case, see, e.g.,
Baxter v. Palmigiano (1976):
“[T]he Fifth Amendment does not forbid adverse inferences against parties to civil actions when they refuse to testify in response to probative evidence offered against them.”
What happens if you take the Fifth in a civil case? [UPDATE: Note important correction as to item 2.]
You can be certain that Trump has good lawyers and he's not worried about incriminating himself during discovery in a civil action. As I recall the criminal charges against Clinton were because he committed perjury, not because of any facts that were uncovered during his deposition. As long as Trump tells the truth, he has no problems. His attorneys will make sure he is not subjected to a fishing expeditions. It is even possible that he will not be deposed or called to testify depending on his personal involvement in the case. But that is another matter for another time.