One important thing that is known about the structure of $SL_2(\mathbb{F}_q[t])$ is Nagao's theorem: for any field $k$, in particular $k=\mathbb{F}_q$, there is an amalgam decomposition
$$
SL_2(k[t])\cong SL_2(k)\ast_{B(k)}B(k[t]),
$$
where $B$ denotes upper triangular matrices. A good reference for statements about the structure of $SL_2(\mathbb{F}_q[t])$ would be Serre's book on trees. For instance, Nagao's theorem can be proved using the action of $SL_2(\mathbb{F}_q[t])$ on the associated Bruhat-Tits tree.

There are differences between the number-field situation and the function field situation, in particular where finiteness properties are concerned. While arithmetic groups in the number field situation do have good finiteness properties, $SL_2(\mathbb{F}_q[t])$ fails to be finitely generated - because of the infinite amalgam summand $B(\mathbb{F}_q[t])$. I think the group also fails to be virtually free for the same reason. (But the reason for this difference to the number field situation seems all concentrated at the prime $p$, somehow.)