1)
EFT Concepts:
1) Why are the lightest heavy quark bound states parity odd?
The answer is pretty simple, and is related to the fact that quarks and antiquarks have opposite intrinsic parity. So for the ground state, a parity odd state is all you have, and this is robust. If you check out other ground state mesons in the PDG, you'll find the same thing. For excited states you start getting (-1)^L type contributions to the parity, where L is orbital angular momentum between the two particles. So the next two heavy meson states (which are L=1 in the quark model) have positive parity.
2) Power counting not in the rest frame
What happens to this discussion and to the power counting rule \partial_t \sim 1/m when we are doing HQET in a frame other than the rest frame, where the leading kinetic energy term is a single gradient term with no m suppression? It sounds like the problem you point out, that p^2/2m is the first term inducing not trivial dynamics, would go away!
Yes, in HQET you do not count \partial_t \sim 1/m. That counting only arises because \partial_t \sim E \sim p^2 / m \sim \nabla^2 / m for Non-Relativistic EFTs of two or more heavy particles. HQET, with a single heavy particle, has a homogeneous scaling for \partial^\mu in frames that are close to the rest frame. You can also look at the power counting of derivatives for HQET in a boosted frame (which is called bHQET). This gives you exactly what you'd expect by applying the boost. See this paper for a description and application of bHQET.
3) What is the most general class of top-down EFTs in which the matching can be accomplished by looking at the coefficients of the 1/\epsilon poles?
In general, it was indicated in class that the theories for which this trick works are precisely those theories for which the low-energy EFT integrals become scaleless.