Field theory Lagrangians are often of the form of a kinetic term plus a source term minus a potential term. How do we know that the potential term is a polynomial in the fields? On a related note why do potentials not include terms that are linear in a field?
On your first question I am not so sure. I would say something like "because its not the kinetic term". The potential term doesn't have linear terms because there would be no vacuum of the theory and the energy would rush/diverge to minus infinity. More generally if you don't have an even power of the field in the "strongest" term of your potential the field theory won't have a (global) vacuum. this means that the energy of your theory could diverge (for some range of positions in the classical theory) and will diverge in QFT (at some point in time your quantum fields will tunnel through the potential barrier). This is unphysical.