"Legal" may be a misleading term in this case.
The "apparent authority" doctrine means that all persons are entitled to rely on the appearance that "Obama" is President.
So, all actions taken by officers of the government, by Members of Congress, and by judges, pursuant to the belief that "Obama" is President would be legal and not reversible.
Does that also go for the continuation of his Presidency (at least until the end of his term)?
No criminal charges?
Personally I think it should be punishable with capital death to lie about eligibility to such an extent. But that's just me, I guess.
I think a lot of us always felt that there would be some recourse, albeit in the future.
Apparent authority looks like it can be used in many other areas as well (bigamy?), but I've never heard of it before, and of course have never heard of it's use.
Even though I do think some things could be based on apparent authority, I think that if tested the final say will be that he had authority once he was sworn in.
The authority for determining if he was eligible was first the electoral college (who seem to have not done a thorough job), and the House (who seemed more than willing to look the other way. Once he satisfied them and took the oath, then he was (qualified or not) president.