(Note that I am only starting to study these works and I may be wrong on some points.)
The paper you are quoting is indeed providing a full definition of (type II and heterotic) superstring theory (type I is missing), valid at the quantum level and for both the NS and R sectors. The definition is basically following the construction of Zwiebach (arxiv:hep-th/9206084) up to two differences:
- In order to treat the R sector (which was the main difficulty up to now) an additional set of free fields (they don't even couple to gravity) are introduced.
- There is no canonical definition of the interactions as one can be found in the paper of Zwiebach (where he is using minimal area metrics) and one needs to go through the world-sheet / off-shell formulation defined in previous papers of Ashoke Sen (arxiv:1408.0571). Doing the same would require to generalize the minimal area prescription to super-Riemann surfaces.
The BV formalism ensures that the action is consistent at the quantum level. The action reproduces the scattering amplitudes (and off-shell correlation functions) and the physical states (plus a set of decoupled free fields) of the usual superstring theories. Due to the fact that it is a field theory loop corrections (susy breaking, vacuum shift, etc.) can be taken into account. Hence there is all the elements that one would require to say that the theory is quantized.
The main (immediate) application is to study some generic properties of (super)string theory in order to show that string field theory (SFT) possesses all the properties to be a well-defined QFT, such as unitarity. Ashoke Sen started this program recently (arxiv:1604.01783, arxiv:1606.03455).
Other goals would be to address non-perturbative effects, but due to the fact that the action is highly complicated (and without intrinsic definition of the vertices) I don't know how far one can go: it is quite inconvenient to work directly with this formulation.
For example the equations of motion are very complicated and it is not really realistic to solve them in order to get solitonic (D-branes) solutions as was done for the cubic bosonic SFT. Moreover I don't think it would help to show the background independence (since this point is not fully understood already for bosonic SFT).
One may hope that it could shed some lights on dualities if one can understand them through the SFT formalism. Hence in my opinion the best we can go for now is to show that string theory is well-defined from the constructive point of view.
To answer your third question I have no clue at all, but this is related to the fact that I have no idea what would be required to prove the adS/CFT conjecture.
Note that a set of 33 lectures given last semester have been recorded, hand-written notes are available (both are available on Ashoke Sen webpage) and we are preparing a review.