Gelfand-Tsetlin algebras and "Jucys-Murphy elements" for $\mathfrak{gl}_n$

1-2) Both of these seem likely, but without doing more reference hunting than I have time for, I won't swear to it. For 2), it's better to think about the $GL_{n-1}$ invariants in $U(\mathfrak{gl}_n)$, which one can understand using the PBW theorem.

3) If you think about the combinatorics here, this just doesn't seem like quite the right framework. The algebra $GT_n$ really has $\binom{n+1}{2}$ algebraically independent generators, whereas the JM elements are just a finite extension of the center (this is clearer if you think about the degenerate affine Hecke algebra). The obvious candidate is the Casimir of the different $\mathfrak{gl}_m$'s, but I don't think this separates the different constituents of the restricted representation in the way you want; I don't think any element of the centralizer will (note that there is a quite interesting connection between the Casimir and JM elements in Schur-Weyl duality, but that's a bit different); it's basically obvious that it can't since the number of generators is just too small. You should just pick your favorite basis of the center of $\mathfrak{gl}_n$; the resulting $\binom{n+1}{2}$ elements will be a free set of generators for $GT_n$. If you were smart in how you chose your basis, then you understand exactly how it acts, based on the GT pattern (if you don't, you need to get a better favorite basis).

4) Sure. What else are you going to consider?