Does the existence of PH-complete problems relativize?
Yao showed, in 1985, that there exist oracles relative to which the Polynomial Hierarchy is infinite. Relative to such an oracle, there don't exist PH-complete problems.

Also, you are right that with a TQBF oracle, PH equals PSPACE. In fact, even P = PSPACE in the presence of a TQBF oracle.