To question 1: One big motivation for me is that two Frobenius algebras can be stable equivalent but not Morita equivalent and a classification up to stable equivalence can be very nice.

For example a classification of representation-finite Frobenius algebras up to Morita equivalence is quite messy (and only known for algebraically closed field, so for simplicitly assume that field are algebraically closed of characteristic 0).

But the classification up to stable equivalence is very nice and can be used to prove things about a large class of algebras by looking only at a much smaller and simpler class of algebras.

For example if you want to prove that the stable endomorphism ring of every indecomposable module of a Brauer tree algebras is isomorphic to $K[x]/(x^n)$ (for some $n$) you can use that all Brauer tree algebras are stable equivalent to symmetric Nakayama algebras where the calculation is almost trivial as even the usual endomorphism rings are isomorphic to $K[x]/(x^n)$ for some $n$. As a bit more non-trivial example you can also determine all stable endomorphism rings or Extension groups (or any other stable invariance) for general representation-finite Frobenius algebras in that way.

Classification results for certain important classes of modules like cluster tilting modules for selfinjective algebras also only depend on the stable module category.

Another example of nice stable equivalence is the rings of simple singularities of type $A_n$, where the stable category of maximal Cohen-Mcaulay modules is stable equivalent to the stable module category of $K[x]/(x^n)$.

Another answer to question 1 might be that certain important functors are only well-defined in the stable module category but not the whole module category, such as the syzygy functor or the Auslander-Reiten translate.

To question 2: To get a "geometric" picture at least for a representation-finite stable category you can take the stable endomorphism ring, which is often a finite dimensional quiver algebra.
For example taking a ring of simple singularities of Dynkin type you will get the preprojective algebra of that Dynkin type as the stable endomorphism ring of the direct sum of all maximal Cohen-Macaulay modules.

To question 3: The stable category of maximal Cohen-Macaulay modules of a Gorenstein ring $R$ is equivalent to the singularity category by a famous result of Buchweitz, see for example in the book https://bookstore.ams.org/view?ProductCode=SURV/262 . When $R$ is Frobenius, then this is just the whole stable module category.