A finite group $G$ all of whose reps are defined over $\mathbb{Z}$ and yet $Rep(G)$ is not generated by permutation representations

The answer is no. Counterexamples include the Weyl groups of types E6, E7, and E8. For a proof that the representations of these Weyl groups are indeed all realisable over $\mathbb{Q}$ (equivalently over $\mathbb{Z}$), see M. Benard, On the Schur Indices of Characters of the Exceptional Weyl Groups, Annals of Mathematics, Vol. 94, No. 1 (Jul., 1971), pp. 89-107 (MSN). For a proof of the statement that $Per(G)\neq Rep(G)$ for these groups, see D. Kletzing, Structure and Representations of Q-Groups, Lecture Notes in Mathematics 1084 (MSN).

The answer is no. For a discussion and counter example, see (for example)
MR3451397 Bartel, Alex; Dokchitser, Tim Rational representations and permutation representations of finite groups. Math. Ann. 364 (2016), no. 1-2, 539–558. https://arxiv.org/abs/1405.6616