For the last question: at least in the algebro-geometric case, the monodromy is always quasi-unipotent (i.e, some power of is unipotent). There is a beautiful argument due to Grothendieck that proves this by reducing to the p-adic case, and using the action of Frobenius on the (tame) inertia group of a p-adic field.

In the general case of a smooth fibration, the only thing I can think of is that, depending on the parity of the degree of cohomology you're looking at, the monodromy ends up living in either a symplectic group or an orthogonal group.

For the last question: at least in the algebro-geometric case, the monodromy is always quasi-unipotent (i.e, some power of is unipotent). There is a beautiful argument due to Grothendieck that proves this by reducing to the p-adic case, and using the action of Frobenius on the (tame) inertia group of a p-adic field.