“I would expect that they would do the most politically expedient thing: declare the office vacant, and then proceed as in the first scenario above with President Biden.”
You may well be right. But if Obama held the office illegally, shouldn’t that invalidate every Executive Order and even some bills he has signed since Jan. 20, 2009? Admittedly, Joe Biden could sign all these bills/orders himself, but in some cases, these bills would have been considered “pocket-vetoed” since Congress recessed during the 10-day period that the Constitution gives the president to sign a bill. That would appear to apply to health care reform reconciliation, since Congress plans to recess for Easter etc.
In my opinion, the executive orders would be invalidated and whomever assumes the office would need to re-institute them (or not, at his discretion).
Legislation, on the other hand, needs to be explicitly vetoed in order to be halted. If the President simply fails to sign it, it becomes law in 10 days anyway. The question becomes how to handle the situation -- the simplest approach is to just say that the bills were unsigned but not vetoed, and are therefore valid law. But that does set a dangerous precedent that Congress can act without Executive branch oversight. It would certainly be ugly either way -- again I would expect expedience to be the main issue. If the new President states that he would have vetoed a particular piece of legislation, it may be an issue -- if he explicitly states that he wouldn't have vetoed anything, then it's just an academic exercise anyway.
No. There is something called the de-facto officer doctrine which says that any acts performed while someone holds office are valid even if it is later found that they are ineligible to hold that office. If Obama is found ineligible and removed tomorrow that would not change anything done to date.