For (1), shtukas are not needed for the construction of automorphic forms from Galois representations. Rather, this is done using the converse theorem. Shtukas are used to check the hypothesis of the converse theorem, involving automorphic forms on lower-rank groups, by converting these to Galois representations. (But shtukas aren't actually needed for this in the rank two case, as you only need the rank 1 Langlands correspondence, which is easier.)

A similar process occurs when we construct the modular form associated to an elliptic curve by taking the $L$-function of the elliptic curve, writing it as a Dirichlet series $\sum_n a_n n^{-s}$, and taking the function $\sum_n a_n q^n$ on the upper half-plane. The work in Wiles' theorem (and Taylor-Wiles and Breuil-Conrad-Diamond-Taylor) is in *proving* this function is a modular form, not in *constructing* it.

The construction of the automorphic form in the function field case is totally analogous to this - one can write down a function using an explicit Whittaker expansion, and the difficulty is in proving that this function satisfies some equivariance property.

The automorphic representation can then be constructed in the usual way from the automorphic form. Alternately, the representation can be constructed "directly" by applying the local Langlands correspondence at each place. Then again the real difficulty is in showing that this representation is indeed automorphic.