About $\Gamma_0$ in the answer cited in the post: it's written that Kruskal's Tree Theorem has an order type "bigger than $\Gamma_0$", which is consistent with the fact that it is the small Veblen ordinal---which is bigger than $\Gamma_0$. The cited paper by Gallier is an expository paper that explains what is $\Gamma_0$, but does not go all the way to the small Veblen ordinal.

For (1), not really: check this answer to a related question, and for more details about the subrecursive stumbingblock see Section 5.1 of Proofs and Computations by Schwichtenberg and Wainer

@Gerhard Paseman: Indeed!

For (2), see en.wikipedia.org/wiki/ELEMENTARY.

Do I understand correctly that this yields $o(P)<\omega_1$ for all $0<\text{rk}\,P<\omega_1$? Couldn't one use the Dushnik-Miller Theorem to a similar end: $\kappa\longrightarrow(\kappa,\aleph_0)^2$ implies $o(P)<|\text{rk}\,P|^+$ for infinite $P$?

Couldn't you bound directly by $o(P)<|P|^+$ in the second paragraph?

@PeterLeFanuLumsdaine true, I understood the question in a restrictive way, as asking whether there existed an ordinal-theoretic function $f$ such that $o(P)=f(h(P))$.

Troesltra's Lectures on Linear Logic.

This type of phenomena has been studied by Andreas Weiermann and several co-authors as phase transitions; e.g. hdl.handle.net/1854/LU-547642 examines when the Ackermann function ceases to be non-primitive recursive. The answer to your question might appear in one of the related papers.