As far as I understand it, by the work of Lafforgue (cf. Laumon, "Cohom. of Drinfeld ... II", Thm 12.4.1) there is a Galois representation associated to an irreducible cuspidal automorphic representation $\pi$ , if $\pi$ is Steinberg at some place $\infty$.

Do we expect Galois representations also if some of these conditions do not hold?

I might be totally off here, but R. Taylor constructed in his thesis (using results from Brylinki-Labesse) Galois rep's to Hilbert modular forms by congruence methods.

Has this be studied anywhere for function fields?

Any hint, where such things are discussed, would be greatly appreciated!