As David says $Q$ is the unipotent radical of $P$. The subgroup $T$ is a preimage of the Weyl group $W$ of the group $G_i\cong GL(V_i/ V_{i+1})$. This group $T$ looks a direct product of $Q$ with a big chunk of a Levi complement of $P$. The Levi complement is a direct product isomorphic to $G_1\times\cdots \times G_k$; to obtain the group $T$, you replace the $i$-th factor by the normalizer $N$ of a maximal split torus $T_0$ of $G_i$.
This is, in fact, the typical way to realize the Weyl group of $G_i$ -- $W$ is isomorphic to the quotient $N/T_0$ -- but this is effectively the same thing as your method of fixing a specific basis of $V_i/V_{i+1}$. The Weyl group rears its head in lots of different ways (most especially as a Coxeter group related to the Dynkin diagram of $G_i$) so this is certainly not the only way to realise it. I don't, however, see any other way to realise your group $T$ (although it depends what you mean by `realise'!).
As for references, it depends on what kind of approach you want. If you want a treatment of $GL_n$ as an algebraic group then I recommend anything by Carter or Humphreys, or else there is the book by Borel. All of these people work in much greater generality than $GL_n$ though. If you just want to understand $GL_n$, then standard algebra texts like the one of Jacobson might be your best bet. (I have e-copies of some of these. If you want them, email me.)