Will Obama stay?

No, Obama can't stay. This is a strange hypothetical, but I would imagine Cruz's VP would take over in such a situation.

But it won't happen. The GOP, after trumpeting themselves for 8+ years as the party that respects the Constitution, is NEVER going to risk running a candidate who could himself be ruled ineligible by the Constitution.

If Cruz wins the nomination (which I doubt) the GOP is going to have to get this legally resolved one way or the other *long* before election day.

In order to file a case with the court you need to have standing

At the Federal level, legal actions cannot be brought simply on the ground that an individual or group is displeased with a government action or law. Federal courts only have constitutional authority to resolve actual disputes (see Case or Controversy). Only those with enough direct stake in an action or law have "standing" to challenge it. A decision that a party does not have sufficient stake to sue will commonly be put in terms of the party's lacking "standing".

Even if someone was successful in filing a lawsuit about Cruz's eligibility it would not be heard before the election and no judge is going to overturn the democratically expressed will of the people.

They REALLY need to start teaching civics in school again. Obama's term in office ends on January 20, 2017. Period. The president elect from the election in November, 2016 is sworn in as president at noon. If that person for any reason cannot assume the office, then the vice president elect is sworn in as VP, and then immediately sworn in as president. If THAT person cannot assume the office of president then the speaker of the house becomes president. That's why the new congress is sworn in a couple of weeks before the new president.

No, Obama is done because he has served his two terms. Whoever Cruz's running mate is will be president if this comes up.

Cruz's eligibility has already been established. By allowing him to run for president, the FEC has already -- at the very least -- tacitly acknowledged that he could take office without incident.

All of the talk about eligibility is just a smoke screen. It won't be an issue.

if Cruz is elected and then found to not be eligible, then his VP will take over

No, whoever the Vice President is will be doing most the work.