If you have power of attorney, there is legally nothing wrong, IMHO, if you're just acting as their agent. If you don't, big problems could come your way.

I'll leave the moral questions to you, but that depends on the circumstances. If it was my Father and it allowed me to visit him more often, I wouldn't consider it immoral.

However logistically, the address on the card should always be their address (wherever they get their other bills). If this is your house, that is fine. If it is not, use what it is.

I have heard of same-address checking account problems, however the big one where you get burned is auto insurance. Never use the same address and auto insurance company as a roomate or you'll soon have an insurer wanting you to add them to your policy.