“Trump” is apparently simply an alteration of the word triumph.
“Trick” on the other hand, I’m not quite as sure of. http://www.dictionary.com gives the following etymology:
Middle English trik, from Old North French trique, from trikier, to deceive, probably from Vulgar Latin *triccare, from Latin tricari, to play tricks, from tricae, tricks.
I suppose the connection could be that deception often plays a part in winning tricks; sometimes it’s tricky to win tricks.
To follow up on Cabbage’s post, he’s correct that trump comes from triumph. What’s interesting is that the Latin triumphus is related to a Greek word (which I don’t know how to reproduce) that means “hymn to Bacchus,” who of course was the God of wine. So there’s an etymological link between drinking and gambling.
The term trick relating to cards goes back to the 1500s. Trick can imply to “skill” in addition to “deception,” so maybe it’s the skill of winning a trick.
As far as kitty goes, the term more properly refers to a pool of money or cards. The original usage was for a pool of money to cover expenses of the game. In poker, it’s the pool of money that goes to the winner of the hand. In other games (e.g. hearts), it is the pool of extra cards that goes to the winner.
I’m not familiar with euchre, but if it is like gin, you are talking about the pile of cards you can pick from each turn. I’ve never heard that pile called the kitty but rather the “stack” or “pile.”
Getting back to the origin of kitty in relation to cards, the OED puts this definition along side the one for the definition of “prison, jail” which comes from kidcote, which as meant “prison” since at least the 1500s. The OED speculates that there was a specific prison named kidcote (or kydcotte), and eventually the name was used to refer to prisons in general. In this sense, the kitty is a pool of money/cards that is temporarily set aside (put in prison).