The latter — there is no certification entity for presidential contenders.
I thought in 2008 it might be the FEC that has responsibility. But they didn’t/don’t. They deferred to the House that certifies state electors. But that is AFTER the election.
The issue has not been addressed. Individual states may have some requirement for a person to get their name on the ballot, but there is no national/Federal entity that certifies the eligibility of presidential candidates.
The court position seems to be — from 2008 — if the people elect him or her, he or she is qualified.
As far as I know, the candidate self-certify. There is no verification
It really isn't FEC business. States run the contests, and have an interest in honest ballots. Opponents can sue each other on eligibility, but in presidential races, no court is going to rule on the merits, because the function of the court in this regard is to subvert the constitution.
At any rate, candidates have to fielded in each state, and the rules vary state by state. I think NH has about 60 (not a type) Republicans running, you can put your own name in, it's easy. Other states restrict the ballot to "candidates offered by qualified parties." Maine is like that. The only GOP names are those that "the party" submits to the SoS. On this front, the GOP in Maine said that it submits the names (and certificates) as directed by the RNC. Technically, the Maine GOP is the entity recognized by the Maine SoS. But if the Maine GOP can't pull a name off if the Maine GOP finds the name ineligible (and that's what I was told), then control over the slate lies with the RNC.
Also, technically, Certification is undertaken personally, by the candidate, in the primary. General election is slightly different - the people are voting for electors, not the party nominees. The Party certifies that it's nominee is eligible, in the general.