There is no sharp distinction between the ways in which physicists and philosophers of physics approach foundational questions in physics. As an example, Carroll's recent paper on self-locating belief and MWI is based on ideas that are heavily pre-figured in the philosophical literature.
I am extremely skeptical about this topic, actually, because there is no way to test it out as I can see, without twisting the definition of testing beyond recognition. Carroll is all like "we derive the Born rule from these reasonable assumptions, therefore MWI", and "it's not an interpretation, it's a formulation", but until he can convince Bohmians or QBists that they are wrong and he is right, I will remain unimpressed.
Many philosophers, especially philosophers of science, are currently engaged in precisely the sort of "hacking away at the edges" problem-solving you endorse. Perhaps you don't see this as a distinctively "philosophical" mode of problem-solving, but that's a semantic quibble.
Absolutely, I don't care how it is called, as long as it is done. I would appreciate a few links to papers which do that, just to understand what you are talking about.
You can't test interpretations. That's why they're called interpretations. It's not that anyone is rejecting empiricism, it is that empirical tests aren't available just because you want them to be.
Why Talk to Philosophers? Part I. by philosopher of science Wayne Myrvold.
See also Sean Carroll's own blog entry, Physicists Should Stop Saying Silly Things about Philosophy.
Sean classifies the disparaging comments physicists make about philosophy as follows: "Roughly speaking, physicists tend to have three different kinds of lazy critiques of philosophy: one that is totally dopey, one that is frustratingly annoying, and one that is deeply depressing". Specifically:
He counters each argument presented.
Personally, I am underwhelmed, since he does not address the point of view that philosophy is great at asking interesting questions but lousy at answering them. Typically, an interesting answer to a philosophical question requires first recasting it in a falsifiable form, so that is becomes a natural science question, be it physics, cognitive sciences, AI research or something else. This is locally known as hacking away at the edges. Philosophical questions don't have philosophical answers.