Basically, the coefficients of an holomorphic cusp form are related to the number of points on a certain smooth projective variety over $\mathbb{F}_p$, and the Weil-Riemann hypothesis gives the neccesary error term for the number of points. This is the content of

The Riemann hypothesis over finite fields was of course proved later by Deligne in his "Weil I" paper.

I'm not sure what the best place to learn these things is (I mean the Ramanujan conjecture, conditionally on the Weil conjecture; for the later there are plenty of resources, see here), other than Deligne's paper, but can find some information here on MO,

As for you third question, there seems to be no relationship between Ramanujan-type conjectures and the location of zeros inside the critical strip, whether you are considering the L-funcion attached to a modular form, or any other L-function of arithmetic or geometric origin.