$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Is the complete representation theory of $\SL(2,\mathbb R)$, $\GL(2,\mathbb R)$, $\SL(2,\mathbb C)$, and $\GL(2,\mathbb C)$ known, in the sense that one can explicitly describe all (nonunitary, projective) irreducible representations, the decomposition of their tensor products, and the structure of all indecomposable reducible representations?
Where would I find the current state of the art?
Yes, this is all completely settled and for the finite dimensional case it is the basic underpinning of much of Lie theory. The irreducible finite-dimensional representations of $\operatorname{SL}(2,\mathbb{R})$ are indexed by the positive integers and can be simply described as $V_m = \operatorname{Sym}^m\mathbb{R}^2$ (no need for projective representations all of these are linear). $\operatorname{SL}(2,\mathbb{C})$ is pretty much identical. None of the finite dimensional representations are unitary. You should find this all in any basic textbook on Lie theory.
The general linear versions are a little more subtle (see here for example) but mostly amounts to taking a $\operatorname{SL}(2,\mathbb{C})$-rep and deciding how the centre acts.
Decomposition of tensor products is again well understood. You can find information under the name Schur-Weyl duality (the 2 dimensional case is sometimes called Clebsch-Gordan theory I believe).
I'm less familiar with what happens with infinite dimensional representations but it is a little more complicated see here
For unitary infinite dimensional representations of $SL(2, \mathbb{R})$ and $SL(2, \mathbb{C})$, I would look at
A. Knapp, Representation theory of semisimple groups, 1986
R. Takahashi, $SL(2, \mathbb{R})$, École d'Été "Analyse harmonique", Université de Nancy I, 1980
S. Lang, $SL(2, \mathbb{R})$, 1985.
