I don't know much about weakly holomorphic modular forms, so what follows is only about holomorphic modular forms. The answer to question 2 is just that this follows from the definition of the period polynomial as an integral and the usual relation between the L function and the Mellin transform.

But the idea is that a cuspidal modular form of weight $k$ and level $1$ is completely determined by its period polynomial, i.e. by the spacial values $L(f,i)$ for $i=1,...,k-1$. In fact you should distinguish the even part and odd part of the period polynomial, each of which characterizes $f$. Zagier (Inventiones Math. 104, 1991) even extended the construction to Eisenstein series (you don't quite get a polynomial, but rather a rational function).

More than that: you get a Hecke equivariant isomorphism between odd period polynomials and cusp forms (you have to be carefull if you want to include Eisenstein series). The action of Hecke operators on period polynomials is explicit. Thus, period polynomials is a useful tools to compute with modular forms, and even study arithmetic properties like congruences between modular forms.

Of course, all this generalizes to higher level. The object you get is usually called ``modular symbols''. I recommend the paper of Loïc Merel (Universal Fourier expansions of modular forms) for modular symbols in the most general setting, in particular for the study of Hecke operators.