Relationships, continue, but can change. Remaining in touch with an ex, is nothing but positive. If current partner cannot see that, it is their problem.
It would depend. If it was from my last ex I would question his intentions. I have a pretty decent friendship with the rest of my exes so I wouldn't be too surprised.
Ebeneezer tried to do me out of his share of the money he promised to pay me for last year's Christmas presents, so I find this question highly amusing.
If it was a freshly broken up couple and there were some unresolved feelings I would use that time to resolve anything and accept the gift. If it was an ex who it has been a long time and we both had committed relationships I would say thanks but no thanks unless we were all friends.
I would probably accept it as not to be rude. I would explain that I'm sorry but I don't have anything for him. I would not want to start a cycle of gift giving, that would be inappropriate and uncomfortable. It would never happen anyway so I have nothing to worry about :-)
I would give them thanks and invite them in for a drink.