If the Senate convicts an impeached officer, the minimum penalty is removal from office. DQ is optional. Alcee Hastings was impeached and removed as a federal judge. Now he's a Congressman, because Congress did not impose a DQ penalty.
Criminal prosecution is discretionary on the part of the prosecutor (note: not on the part of the FBI!), for everybody. Hastings was tried for the crime, criminally, before he was impeached. Ion other words, the fact that he held office did not prevent him from being charged, criminally, and being subjected to a criminal trial.
The president holds a unique position - there is only one. Hence the academic interest in whether he can be indicted while in office, or if that has to wait until after he is out of office, by hook or by passage of time.
Impeachment doesn't have any conflict of interest consideration. Impeachment is deliberately a biased, partisan activity. Whatever justification is brought up, is done to convince the people that the target is a skunk and ought to be removed from office for that. Removed for misdemeanors. The target of impeachment always tries to turn it into a legal standard, find the statute, etc., but if enough members of Congress want to impeach and remove somebody for bad breath, there is no recourse. As noted, the throttle on this power is accountability to the electorate, plus limited penalty.
The issue here is the investigation leading to impeachment (formal charges).
The investigation should be done probably by the House IMO and not the DOJ because of prima facia conflict of interest.