Number of equivalence classes in regular languages as a function of DFA size
Answer to Question 1,


  What is the smallest $n$ for which $|T/{\sim}|=|T|$?


We have $$n=\max_{|w|=|x|=s,\\ w\ne x}\mathrm{sep}(w,x)$$
where $\mathrm{sep}(w,x)$ is the smallest number of states in any DFA that accepts one of $w$ and $x$, but not the other.
The best known upper bound on $n$ is then (see some slides by Jeffrey Shallit)


  $n=O(s^{2/5}(\log s)^{3/5})$.


which was obtained in

Robson, J. M., Separating strings with small automata, Inf. Process. Lett. 30, No. 4, 209-214 (1989). ZBL0666.68051.