Relaxation of the null production restriction in Regular and Context Free Grammars
True.

For CFG we can remove in fact all epsilon rules and obtain the same family of languages, except for the empty word. That can be added by adding the $S\to\varepsilon$ production. It is a so-called normal form.

Like Raphael says: it all depends on what you want to express in the formalism, and what makes proofs convenient.

The definition on type-3 grammars is a matter of taste. I would prefer $A\to\varepsilon$, $A\to aB$ as these two types of productions correspond to final states and transitions in finite state automata.