This is a brief answer; possibly others have different opinions about this.

**Question 1**: The Langlands conjectures gives a correspondence between Galois representations and automorphic forms. So a very naive way to view the zeta function of a variety over a finite fields from this philosophy is to look for a Galois representation which it comes from. However, this is exactly the point of the Weil conjectures; the zeta function has a description in terms of the action of the absolute Galois group of the finite field on the etale cohomology of the variety. These indeed give the Euler factors.

**Question 2**: The Riemann hypothesis for the Weil conjectures over finite fields corresponds to the *Ramanjuan conjecture for automorphic forms*. In fact, this was one of Deligne's original applications of the Weil conjectures, to proving the Ramanjuan conjecture for the Ramanjuan tau function.

**Question 3**: The answer is that it is a mixture of 1) and 2). For 1), as I said above, the Weil conjectures give you results towards the Ramanjuan conjecture for automorphic forms. For 2), it is a standard result that each Euler factor of an automorphic $L$-function is a rational function in $p^{s}$, but proving the rationality of the zeta functions of varieties over finite fields was one of the first difficult steps in the proof of the Weil conjectures.