Thanks! So it seems the question has a simple answer in both the finite and Lie cases, but the answer is very different in these two cases.

Does "degree" mean the same thing as dimension of a representation? In which case are you saying that if $r_2$ is an ordinary representation then the answer is always "yes" to the second question, as I can tensor in $T_1$ and $T_2$ to make the dimension of $r_1 \otimes T_1$ match that of $r_2 \otimes T_2$ and then pick $R$ to be the regular representation?