I always thought we didn't know if the universe is finite or infinite. However is it true that the universe (not just the observable one) is finite with no boundaries?
Also, is it true that in terms of quantum mechanics there is and should be a meaning to the term "outside the universe"?
You are right that we do not know if the universe is finite or infinite in space. Cosmologists do now think that it has an infinite future because of the accelerated expansion rate due to dark energy but this does not tell us anything about the question of infinite space.
To answer the question for space we first have to assume spacial homogeniety, i.e. that space looks the same everywhere on large scales. If this assumption is wrong then we are stuck because we cannot see beyond the horizon of the observable universe which is due to the finite speed of light and the finite age of the universe. If we can't assume anything about what happens beyond the horizon then obviously we cant tell if it is finite or infinite.
However, within the observable universe space does appear to be homogeneous so it is usual to assume, rightly or wrongly, that it is homogeneous everywhere. If that is the case then the question of the finiteness depends on the curvature of space and this is something that can be estimated with some precision using cosmological observations, especially that of the cosmic microwave background.
If the curvature of space takes a positive value then it must be finite. If it is zero or negative then it is probably infinite though the possibility of a finite universe remains if it has an unusual topology like a tessellation of a finite polyhedron.
When we measure the curvature we find that it is close to zero and we cannot tell whether it is positive, zero or negative with the present error bars, so we do not know if the universe is finite or infinite in space. This flatness is expected as a prediction of inflation theory.
The best we can do is set a lower limit to how small the universe can be given limits on its measured curvature and that is what the papers you linked to are trying to do. They do not claim to settle the question of whether it is finite or infinite.
Your final question about quantum mechanics and the outside of the universe is unrelated and should have been asked as a separate question, but the answer is no.