*Disclaimer: I'm far from an expert on any of the topics of this question. I apologize in advance for any horrible mistakes and/or inaccuracies I have made and I hope that the spirit of the question will still be clear despite them.*

The (integral) representation rings of the symmetric groups can be packed together into a hopf algebra $H_1 = \oplus_n Rep(\Sigma_n)$ where the multiplication (resp. comultiplication) comes from induction (resp. restriction) along $\Sigma_n \times \Sigma_k \to \Sigma_{n+k}$. In fact there's a further structure one can put on $H$ corresponding to the inner product of characters and a notion of positivity (all together its sometimes called a "positive self adjoint hopf algebra"), but for simplicity I will disregard this structure in what follows (of course if its not important for the answer that would be great to know).

Its well known that sending the irreducible specht modules to their corresponding schur functions induces an isomorphism of hopf algebras to the (integral) hopf algebra of symmetric functions.

Following the "$\mathbb{F}_1$-philosophy" it is tempting to define a ring of "q-symmetric functions" as the hopf algebra $H_q = \oplus_n Rep(GL_n(\mathbb{F}_q))$ equipped with the same structures as above.

Question 1:Is there a hopf algebra over $\mathbb{Z}[q]$ which specializes at a prime power $q=p^n$ to $H_{p^n}$ and at $q=1$ to $H_1$ the classical ring of symmetric functions?

By schur weyl duality we also know that $H_1 \cong Rep(GL_{\infty}(\mathbb{C})):= colim_n Rep(GL_n(\mathbb{C}))$ (at least as rings). It seems natural to ask if there's any form of schur duality going in the other direction.

Question 2:Is there any kind of relationship between the rings $Rep(\Sigma_{\infty}) := colim_n Rep(\Sigma_n)$ and $\oplus_n Rep(GL_n(\mathbb{C}))$?

Question 3:Is there a $\mathbb{Z}[q]$-algebra which specializes to $Rep(GL_{\infty}(\mathbb{F}_q))$ at a prime power $q = p^n$ and to $Rep(\Sigma_{\infty})$ at $q=1$?