EFT Concepts:

1) Power counting not in the rest frame

What happens to this discussion and to the power counting rule  $//<![CDATA[ \begin{array}{l}\partial_t \sim 1/m\end{array} //]]>$ 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  $//<![CDATA[ \begin{array}{l}p^2/2m\end{array} //]]>$ is the first term inducing not trivial dynamics, would go away!

Yes, in HQET you do not count  $//<![CDATA[ \begin{array}{l}\partial_t \sim 1/m\end{array} //]]>$. That counting only arises because  $//<![CDATA[ \begin{array}{l}\partial_t \sim E \sim p^2 / m \sim \nabla^2 / m\end{array} //]]>$ for Non-Relativistic EFTs of two or more heavy particles. HQET, with a single heavy particle, has a homogeneous scaling for  $//<![CDATA[ \begin{array}{l}\partial^\mu\end{array} //]]>$ 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.