2) Was a person born in “the Republic of Texas” (1836-1846) eligible for election to the presidency sometime after Texas is admitted to the Union?
The closest example I can think of that comes close to this situation is Barry Goldwater being born in Phoenix before it was a state. However, it was considered a territory (As originally declared by Jefferson Davis and subsequently accepted after the civil war). Not exactly the same situation but, there doesn't seem to be much precedent to compare with. Suffice to say, I have no definite answer.
First off, other than having the same name, the Confederate Territory of Arizona had little to do with the later Federal creation. It didn't even have the same boundaries. Confederate Arizona was the half of the then Territory of New Mexico south of 34 N. The Federal Territory of Arizona (first proposed in 1858) was the half of New Mexico west of the current Arizona/New Mexico border.
Secondly, as I pointed out already, Vice President Curtis is a far stronger example for determining whether a person born in U.S. territory that is not yet a state is a natural-born citizen than Goldwater ever would be. He was not merely a candidate, Charles Curtis actually served in a position that had the natural-born citizen requirement.
Thanks for the history lesson