Does a DPOA cover all finances (different bank accounts) or is a separate POA needed for each account?

If your DPOA specifies that you will take financial decisions on your aunt's behalf, then yes you can. Some banks are more up to speed on POA than others, you may have to have patience while they scratch their heads and read it with their lips moving; and they may insist on seeing an original; but you should get there in the end.

Yes, you can give them a copy. Do you live in a county where Durable POAs are filed with the clerk of court? The reason that I ask is if they do, then than would add a stamp of authenticity to it and if you are ever asked for a certified copy, they would be able to give you one for a small fee of course.

Bring the original to the bank, not a copy. They can make their own copy.

Do you have conformed copies? They can stand in the place of an original but have the same legality. That's why good attorneys have 2 - 3 copies executed, with those being the conformed copies.