How does string theory get an agreement to the Hawking-Bekenstein equations to calculate the quantum entropy of a black hole in a background-dependent way?
Is there any sort of area parameter in string theory like Barbero-Immirzi in loop quantum gravity?
I personally recommend reading the original Vafa-Strominger computation. In particular the example of the entropy computation of a black hole in type IIA string theory on $K3$ (chapter 4.1 https://arxiv.org/abs/hep-th/9702201). That example simply and beautifully encodes the fundamental "trick" that makes black hole thermodynamics understandable (when possible) within String/M-theory.
$D$-branes can be described by means of the physics of the open strings that are attached to them and black holes can be considered as ensembles of D-brane bound states in the context of string theory; then the strategy for the computation is to compute oscillator degeneracies of strings describing the effective theory of large $D$-brane charge ensembles.
The observation is that typical indexed black hole degeneracies of $D-$brane charge end up being reduced to the study of the growth of the cohomology of $n$-symmetric products of the internal space. Or more accurately, to integrations over $n$-Hilbert Schemes of points. Amazingly the latter problem can be studied by analyzing sigma-models with the $n$-symmetric product as target spaces. That is the beauty of string theory, the asymptotic Cardy-growth of states in an RCFT (describing a string propagating in a symmetric product of the internal space) organize the growth in the cohomology of moduli spaces of $n$-symmetric products of the extra dimensions. That's also why string theory is so powerful in predicting new phenomena in enumerative algebraic geometry.
In our case (type IIA string theory on an elliptically fibered $K3$) begin by fixing a second homology element of $K3$ and an integer number $n$. To compute the $D-$brane degeneracies for that homology representative at level $n$ you wrap $n$ $D2$-branes on that homology class and a 1-cycle on the elliptic fiber. Then $T$-dualize that 1-cycle, the resulting configuration is equivalent to the choice of a point in the fiber; then every choice of such a point defines a point in the moduli space of $n$ $D2$-branes wrapped on a fixed homology class, by varying that point over the fiber, you can recognize why the $n$-fold symmetric product of $K3$ is the space of possible microstates of a black hole with $n$ $D2-$brane charge, taking the cohomology of the space identifies the microstates up to exact BRTS-cohomology classes.
Observation: The computation of the black hole entropy in string theory can be achieved also in a background-independent way. That can be seen in many ways, I just want to mention two:
The topological string $B$-model can be used to compute the entropy of extremal black holes (see Black Hole Attractors and the Topological String ). The physics of the spacetime description of the $B$-model closed string can be completely computed by the holomorphic anomaly equations (references 1 and 2) that give a workable description of what "background-independence" means in quantum gravity/string theory.
If you are interested in computations of the entropy of black holes by means of "counting quanta of spacetime" in string theory. I recommend Crystal Melting and Black Holes based on the quantum foam description of topological string theory Quantum Foam and Topological Strings . Here the strategy is precisely to vary the background (by adding chern-classes) and counting certain sheaves over each choice. After summing over all possible geometries, the black hole entropy is exactly computed, including prefactors.
To answer your second question: No, there are no Barbero-Immirzi-like parameters in string theory because in string theory every parameter is the VeV of a scalar field. Free parameters contradict the spirit of string theory.
References:
Conceptual Analysis of Black Hole Entropy in String Theory.
Strominger and Vafa derived the Bekenstein-Hawking entropy from counting degenerate vacua in string theory:
https://arxiv.org/abs/hep-th/9601029
I don't know the answer to the background independence or to the loop QG thing. Note that this paper is from before the discovery of AdS/CFT.
