Algorithm to determine a set given the size of its intersection with sets you choose
You are asking two questions. I will only answer the first, ignoring the fact that $|T|$ is much smaller than $|U|$. Your problem is very similar to the one tackled in this question. The same lower bound of $\Omega(n/\log n)$ holds here as well, where $n = |U|$. The same (apparently) matching upper bound also holds, since you can simulate a Hamming distance query using your kind of query.