This is a quick answer to explain the statement that the hard direction of Schur-Weyl duality is the same thing the First Fundamental Theorem of invariant theory.
Let $V$ be a finite dimensional vector space and $V^{\ast}$ the dual space. The FFT (or a special case there of) says that the $GL(V)$ invariant multilinear functions $V^n \times (V^{\ast})^n$ are spanned by the functions
$$(v_1, v_2, \ldots, v_n, v^{\ast}_1, v^{\ast}_2, \ldots, v^{\ast}_n) = \prod \langle v_i, v^{\ast}_{\sigma(i)} \rangle $$
for $\sigma \in S_n$.
Translating "multilinear functions" into "tensors", this gives a spanning set for the $GL(V)$ invariants in $V^{\otimes n} \otimes (V^{\ast})^{\otimes n} \cong \mathrm{Hom}(V^{\otimes n}, V^{\otimes n})$.
Unwinding the isomorphism $V^{\otimes n} \otimes (V^{\ast})^{\otimes n} \cong \mathrm{Hom}(V^{\otimes n}, V^{\otimes n})$, we see that the $GL(V)$ action on $\mathrm{Hom}(V^{\otimes n}, V^{\otimes n})$ is conjugation by the $GL(V)$ action on $V^{\otimes n}$. So a $GL(V)$-invariant is precisely a map $\mathrm{Hom}(V^{\otimes n}, V^{\otimes n})$ which commutes with the $GL(V)$ action. And the $n!$ spanning elements found above turn into the $n!$ maps which permute tensor factors. So FFT says that a map $V^{\otimes n} \to V^{\otimes n}$ which commutes with the $GL(V)$ action is a linear combination of permutation of tensor factors, just as you want.
Yes. In combinatorics this is known as Robinson-Schensted-Knuth vs. just Robinson-Schensted. (Properly speaking the latter is about a yet smaller duality, $\mathbb C[S_n] = \bigoplus_{\lambda\vdash n} \sigma_\lambda \otimes \sigma_\lambda^*$.)
First, shrink the Peter-Weyl result from $\mathcal O(GL(n))$ (you overuse $V$, I feel) to the slightly smaller $\mathcal O(M_n)$. Then the RHS shrinks to $\oplus_\lambda V_\lambda \otimes V_\lambda^*$, where $\lambda$ now runs over partitions $(\lambda_1 \geq \ldots \geq \lambda_n \geq 0)$ instead of all dominant weights $(\lambda_1 \geq \ldots \geq \lambda_n)$.
Then generalize to other matrix spaces, not just square matrices, obtaining $\mathcal O(M_{a\times b}) \cong \bigoplus_\lambda V^a_\lambda \otimes (V^b_\lambda)^*$, the sum now over partitions of height $\leq \min(a,b)$.
(The combinatorial statement, RSK, is a bijective proof of two different character formulae for this representation. The obvious weight basis is given by monomials in the matrix entries, equivalently listed as $M_{a\times b}(\mathbb N)$. On the RHS we have pairs of same-shape SSYT. Under the bijection the row and column sums of the matrix in $M_{a\times b}(\mathbb N)$ go to the contents, i.e. entry multiplicities, of the two SSYT.)
Now, consider functions on $M_{a\times b}$ of weight $(1,1,\ldots,1)$ under the $T^a \leq GL(a)$ action. Since that's $S_a$-invariant and $S_a$ normalizes $T^a$, this weight space will have a $S_a \times GL(b)$ action.
The LHS will be made of functions that are multilinear in the rows, i.e. $(\mathbb C^b)^{\otimes a}$. The representation $V^a_\lambda$ has a $(1,1,\ldots,1)$ weight space iff $\lambda$ is a partition of $a$, and in that case, the $S_a$ action on it is the Specht irrep $\sigma_\lambda$ of $S_a$. Which is to say, the RHS has become $\oplus_{\lambda \vdash a} \sigma_\lambda \otimes (V_\lambda)^*$ like you wanted. QED.
(Now we're insisting that the row sums are all $1$. On the RHS, one of the SSYT is an SYT. If you go further and ask that the column sums be all $1$ also, then the LHS becomes just permutation matrices, the RHS pairs of same-shape SYT, and the correspondence is just Robinson-Schensted no Knuth.)
As I recently learned from Martin Kassabov, you can run this in reverse: take two copies of the Schur-Weyl isomorphism, reverse one, and tensor them together over $\mathbb C[S_n]$ to get the Peter-Weyl (for matrices) result. So it's a matter of taste deciding which one is the more fundamental.
Best Answer
The questions here have certainly been explored (though not definitively) in many recent papers or preprints on arXiv. Look for example at the arXiv paper by Stephen Doty Link, as well as many others by Steve and/or his collaborators. Most of the arXiv papers have subject listing RT (some also consider quantum analogues under QA). But some predate arXiv; there has been a lot of study of decomposition numbers of symmetric groups in prime characteristic, for example, using what little is known about modular representations of GL$_n$. Not having gone far with this literature myself, I'd suggest that you start the inquiry with available papers and then maybe raise narrower questions here.
My main point at first has been that a lot of literature exists from the past couple of decades, so the questions should focus on what is in that literature (not just Doty's short conference paper I cited). Concerning the status of modular representations for finite groups of Lie type, what's known is not yet good enough to answer most questions about symmetric groups for small primes. While Lusztig's conjectures promise a good conceptual picture for primes at least the Coxeter number of the Weyl group, even that much can be implemented only in recursive style. For small primes little is known, but it would have immediate applications to $S_n$. In classical Schur-Weyl duality the dictionary goes the opposite way.