To steal a cheesy movie line from my youth... "A strange game. The only winning move is not to play. How about a nice game of chess?"
She is playing with you. Of course it is inappropriate. And the most inappropriate part is that she is putting your 11 year old son in the middle of this little game. You shouldn't play.
My advice is to ignore it. You have no obligation to have anything to do for your ex-wife's birthday. There is no reason that you need to be involved in your son's gift to your ex-wife.
I wouldn't even respond to these emails. I would just ignore them. If I felt obligated to get something... it would be something innocuous (maybe a toaster).