String theory leads to the falsifiable claims of gravity and quantum mechanics. I don't know why this non-falsifiability argument keeps getting repeated! It would be nice for string theory to make more claims than just that, but they're still working on it. LQG is an example of a framework that truly makes no claims, because there isn't even a LQG setup for three dimensions of space. To my knowledge it hasn't even been proven that there is an LQG setup for three dimensions of space.
Yes. It has "falsifiable claims" in the sense that, if existing theories of e.g. general relativity and quantum mechanics were proven false, it would also be proven false. Generally when people refer to "falsifiable claims" they're referring to falsifiable claims that are not already made by a more parsimonious theory. As far as I understand the field, string theory has made no such claims.
And yes, that's what I meant by LQG being a less mature theoretical space. Either LQG will fail to find a solution for 3D space (so sad), or it will and its solution will make falsifiable predictions that are not part of existing theory (yay!), or it will propose a set of solutions that make no new falsifiable predictions (in which case it enters the same state as string theory). Both of those first two possibilities would be forward progress.
