Not an expert but my comments are too long to fit in the comment box:
As far as I know, the "coefficients" are a well-behaved (e.g. triangulated) category with some rich structures, namely six operations à la Grothendieck:
pull-back, push-forward, tensor product, inner himhom, upper and lower shriek.
They are expected to satisfy various functoriality and adjunctions, which are similar to the case of sheaves of abelian groups on a topological space.
These structures at least allow you to connect coefficients on different spaces, and thus do dévissage. Once the category of coefficients are understood well, some hard theorem can be reduce to the curve case, where you still need to work hard... (Hopefully this partly answers question 1.)
Also, one Weil cohomology theory you didn't mention is crystalline cohomology, which coincide with rigid cohomology for proper smooth varieties. There, as I read from Illusie's survey, a satisfactory category of coefficients is missing. I don't know if there is any further development after that.