The answer by @orangeskid (+1) is the most classical and direct, and the answer by @Hanno connects your question to Schubert calculus and the geometry of flag varieties (+1). Hoping this isn't too self-promoting, you might also have a look at my paper Jack polynomials and the coinvariant ring of $G(r,p,n)$ (I worked in somewhat more generality)
where I showed that certain non-symmetric Jack polynomials give a basis as well (I was working with more general reflection groups and over the complex numbers, but a version should work over any field of characteristic $0$; I have not thought much about this in characteristic $p$). The point of my paper was really to connect the descent bases (yet another basis!) studied much earlier by Adriano Garsia and Dennis Stanton in the paper Group actions of Stanley-Reisner rings and invariants of permutation groups
to the representation theoretic structure of the coinvariant algebra as an irreducible module for the rational Cherednik algebra. Of course this structure becomes much more complicated in characteristic smaller than $n$; in particular the coinvariant algebra will in general not be irreducible as a module for the Cherednik algebra.
Towards your second question: by definition $p$ is flat if and only if $k[V]$ is a flat $k[V]^G$-module. This certainly holds if $k[V]$ is free over $k[V]^G$; a sufficient condition for this is given in Bourbaki, Theorem 1 of section 2 of Chapter 5 of Lie Groups and Lie algebras (page 110): in case the characteristic of $k$ does not divide the order of $G$, it suffices that $G$ be generated by reflections. This is false (in general) in characteristic dividing the order of the group. But see the paper Extending the coinvariant theorems of Chevalley, Shephard-Todd, Mitchell, and Springer by Broer, Reiner, Smith, and Webb available for instance on Peter Webb's homepage here
for references and what can be said in this generality (this is an active area of research so you shouldn't expect to find a clean answer to your question).
Conversely, assuming $p$ is flat and examining the proof of the above theorem in Bourbaki, it follows that $k[V]$ is a free $k[V]^G$-module. Now Remark 2 to Theorem 4 (page 120) shows that $G$ is generated by reflections. So to sum up: if $p$ is flat then $G$ is generated by reflections; in characteristic not dividing the order of $G$ the converse holds.