These are just some random remarks, with one hopefully useful reference.

"Are there some cases where the $f$-vector specifies completely the polytope?"

This is hardly what you are seeking, but for 3-polytopes, $f_2=2f_0-4$ is achieved exactly for the $f$-vectors of simplicial polytopes. And of course the stacked and cyclic polytopes achieve the lower and upper bounds respectively.

I don't know if you have seen Günter M. Ziegler's "Convex Polytopes: Extremal Constructions and $f$-Vector Shapes" (IAS/Park City Mathematics Series Volume 14, 2004), which seems to directly address your questions, albeit as of several years ago. Here is the PDF.

Here is one tidbit. He mentions, as a measure of our ignorance, that not even this "suspiciously innocuous conjecture" of Imre Bárány is settled:

For any $d$-polytope, $f_k \ge \min \{f_0, f_{d−1}\}$.

It is (or was in 2004) only proven for $d \le 6$.

Günter has a particularly careful description of what's known about the $f$-vectors of 4-polytopes, a specialty of his. In particular, the set of these $f$-vectors "is not the set of all integral points in a polyhedral cone, or even in a convex set." It has concavities and holes.

These are just some random remarks, with one hopefully useful reference.

"Are there some cases where the $f$-vector specifies completely the polytope?"

This is hardly what you are seeking, but for 3-polytopes, $f_2=2f_0-4$ is achieved exactly for the $f$-vectors of simplicial polytopes. And of course the stacked and cyclic polytopes achieve the lower and upper bounds respectively.

I don't know if you have seen Günter M. Ziegler's "Convex Polytopes: Extremal Constructions and $f$-Vector Shapes" (IAS/Park City Mathematics Series Volume 14, 2004), which seems to directly address your questions, albeit as of several years ago. Here is the PDF.

Here is one tidbit. He mentions, as a measure of our ignorance, that not even this "suspiciously innocuous conjecture" of Imre Bárány is settled:

For any $d$-polytope, $f_k \ge \min \{f_0, f_{d−1}\}$.

It is (or was in 2004) only proven for $d \le 6$.

Günter has a particularly careful description of what's known about the $f$-vectors of 4-polytopes, a specialty of his. In particular, the set of these $f$-vectors "is not the set of all integral points in a polyhedral cone, or even in a convex set." It has concavities and holes.