I'm trying to figure out/find in literature the details concerning Gelfand-Tsetlin algebras for $\mathfrak{gl}_n(\mathbb C)$ (Okounkov-Vershik style, if you wish).

Consider the chain $$\mathcal U(\mathfrak{gl}_1)\subset\ldots\subset\mathcal U(\mathfrak{gl}_n),$$ denote the centers $Z_i\subset\mathcal U(\mathfrak{gl}_i)$ and define the commutative algebra $$GT_n=\langle Z_1,\ldots,Z_n\rangle.$$

One sees that elements of $GT_n$ act diagonally in the Gelfand-Tsetlin basis in any finite-dimensional irrep of $\mathfrak{gl}_n$. Beyond that little is clear to me. It'd be great if someone could provide an answer or reference for any of the following. (Of course, 3) is the key question here.)

1) Is $GT_n$ *precisely* the subalgebra of $\mathcal U(\mathfrak{gl}_n)$ acting diagonally in any GT basis? Is it a maximal commutative subalgebra?

2) Is the centralizer $Z(\mathcal U(\mathfrak{gl}_n),\mathcal U(\mathfrak{gl}_{n-1}))$ commutative? Do we have $$Z(\mathcal U(\mathfrak{gl}_n),\mathcal U(\mathfrak{gl}_{n-1}))=\langle Z_{n-1},Z_n\rangle?$$

3) Are there generators of $GT_n$ analogous in some way to the Jucys-Murphy elements? How do they act on an element of a GT basis in terms of the corresponding GT pattern (or SSYT)?

4) Is $GT_n$ at all the natural object to consider here?