Why [is BOP not a principle we should always endorse]?
That's tricky. If we accept BOP, then the BOP is on you to show why we should endorse it, since you are the one making that claim. But this is dumb. I (and anyone working in epistemology) simply disagree that we should even accept it as a principle of good reasoning, since there is no independent epistemic reason to do so. BOP e.g. rules out self-evident propositions (e.g. why P follows from the truth of P&Q), and it let's people off the hook, who really ought to justify their beliefs, like certain atheists. Beyond that, I'm going to be repeating myself here, but "burden of proof (BOP) isn't a principle in logic. It's a rule of thumb, which is e.g. very useful in a court of law.". I can't explain why it isn't a principle in logic. It simply isn't. I also can't explain why it isn't a type of fish. Open any introductory book on logic to see for yourself. Beyond that, it isn't a universally accepted principle in epistemology either. Because you always have to justify your position or beliefs, no matter whether you asserted them or someone else did. Any belief needs a justification, unless it's simply blind faith. So the atheist needs to justify why god don't real, if that is what they really believe.
No. The theist is making the claim. The atheist is not. The burden of proof reside with the one making the claim.
Both of them are making claims. How is "god does not exist" not a claim?
What theories? That I reject hard solipsism? OK. I reject it. OR more precisely -- I don't care. Why? Because a difference that makes no difference is no difference. If you claim that a world with a deity behaves exactly the same way as a world without a deity, then congratulations. We're debating tautology -- a singularly uninteresting subject matter.
What theories? The one's I mentioned! But there are probably many theories you accept which aren't falsifiable. I didn't - AFAIK - mention solipsism, I mentioned materialism and causation. Further, what you care about doesn't really come into it. If you hold yourself to a certain epistemic standard, it seems odd that you would hold someone else to a higher standard, and reject their views in light of that standard.
In general, nothing. In specifics, everything. Something as monumental as a deity -- particularly a supernatural one -- justifies the most rigorous scrutiny imaginable. It deserves nothing less.
O.K. Why?
No. Something is wrong with your theory of truth.
Falsification is not a requirement for something to be true. It is - arguably - a requirement of something to be a good scientific theory. 2+2 = 4 is e.g. not falsifiable, but it's obviously true.