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 of a maximal split torus of $G_i$.

This is, in fact, the typical way to realize the Weyl group of $G_i$: $W$ is the quotient $N/T_0$ where $N$ is the normalizer of a maximal split torus $T_0$ in $G_i$ - 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 realize your group $T$.

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.)