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.