No, one cannot prove a negative, unless the parameters are very narrow and can be fully investigated - however this is not relevant.

The burden of proof is always on the person making a claim. It's not up to everyone else to prove you wrong. It's impractical. There are limitless possibilities that haven't been proven to be wrong yet. That doesn't mean you go ahead and accept all of them until they're disproven.

Many theistic claims aren't constructed in such a way that they're falsifiable. That makes them automatically non-credible.

As a general rule of thumb, if an argument you are using, equally justifies an opposing viewpoint, then it's not a great argument. For instance, if I were to ask a Christian, "Can you prove that Allah doesn't exist?", he/she would scoff at the argument. Asking a Muslim, "Can you prove that Jesus-God doesn't exist?", he/she would equally scoff at the argument. Both, however, use the argument as a justification for their own religions.

Now, determine the permutations of 7 major religions asking each other that question, and you'll find that none of them find the argument convincing from any of the other religions' positions.

That's because the process of accepting claims as true until they're falsified is an invalid and inaccurate approach to gathering knowledge.