I have some questions about the interplay of interpretability, model theory and category theory. Since I had difficulties in finding literature or other helpful information about this topic, it would be great if somebody of you can help me.

For a first-order theory $T$, let $Mod(T)$ denote the category of all models of $T$ where the arrows are the homomorphisms (or - if you want so - take the elementary embeddings). Let $I_T$ denote the spectrum of $T$. That is, $I_T$ is a function which takes some cardinal $\kappa$ as argument and which outputs the number of non-isomorphic models of $T$ having cardinality $\kappa$. I will also refer to the syntactical notion of interpretability (i.e. "theory $T_1$ interprets theory $T_2$") and mutual interpretability ("$T_1$ interprets $T_2$ and vice versa"). For what follows, assume that $T_1$ and $T_2$ are arbitrary first-order theories. Here are the three questions for which I would like to know whether there are any answers:

Are $Mod(T_1)$ and $Mod(T_2)$ equivalent (in the category theoretical sense) whenever $T_1$ and $T_2$ are mutually interpretable? Does the converse holds?

Are $Mod(T_1)$ and $Mod(T_2)$ equivalent (in the category theoretical sense) whenever $I_{T_1}=I_{T_2}$? Does the converse holds?

Does $I_{T_1}=I_{T_2}$ holds whenever $T_1$ and $T_2$ are mutually interpretable? Does the converse holds?

I will be grateful for any comments, answers and also for references to literature (papers, books,...) in which these questions or similar ones are discussed. Thank you!

Note that I do not expect that these questions are "profound conjectures" or something like that. Maybe that among the experts, their answers are already known and even "trivial". I was just thinking about some model theoretical topics and in this context, the three questions from above started to occupy me. Since I am not able to find any literature concerned with similar questions, I just would like to know whether there are some people in this forum who know more than I do and could share their knowledge with me.

**Edit**: Following some remarks of some of the commentators of this post, the above questions may also be interesting if we replace "mutual interpretable" with "bi-interpretable".