To add to JT's remark, we also need that when $f_1:E\to F_1$ and $f_2:E\to F_2$ are isogenies of elliptic curves with $f_1$ separable and $\ker f_1\subseteq\ker f_2$ then there is an isogeny $g:F_1\to F_2$ with $f_2=gf_1$. Over $\mathbb{C}$ where elliptic curves are complex tori, this is quite easy to prove. Over general fields it requires more work; see Silverman's book for instance.

With no extra effort, you could reduce your bound to $m|U|^n$. The argument in 2) gives an upper bound of $(m-1)|U|^n$ which you add to the $|U|^n$ from 1).

It certainly comes into Greg Kuperberg's proof of the Alternating Sign Matrix conjecture: Another proof of the alternating-sign matrix conjecture, Internat. Math. Res. Notices (1996), 139-150.

Thanks, Kevin, I'll need to bone up on integer points :-)

If $V$ is the representation space, then the space $T$ corresponds to $G$-maps from $\mathrm{Sym}^2(V)$ to $k$ considered as a trivial $G$-space. Thus $\dim T$ is the number of copies of the trivial representation inside $\mathrm{Sym}^2(V)$. This won't change when one passes from $k$ to an extension field.