"I like word games" was accepted as correct. If we wanted to mention word games (like Scrabble), would we use jeux de mots or is there another way to say word games?
Disclaimer - not a native speaker, French is my second language. I believe for games like Scrabble, "jeux de mots" is acceptable, as are "jeux de lettres", and "jeux en ligne". The last one is particular to a Scrabble/crossword type game.
des, can also be a proposition. Why the proposition de is used instead of the proposition des is due to the fact that the noun that follows the proposition is unspecific (if it was specific des would be correct). More information at http://french.about.com/od/grammar/a/de-vs-du-de-la-des_3.htm
Games of words is wrong. I don't have a rule/link for it. That's just the way it is in English.
Games involving words = word games
Games involving balls = ball games
Games involving boards = board games
Since I don't like all puns, but only some of them, would I say, "J'aime certains jeux de mots"? Or J'aime des jeux de mots"?
This was already answered in this discussion thread. See my reply to evelim1972.
In the future, please read the threads before posting questions.
"i like word play" was marked as incorrect but we do not say it in plural in English 'i like word plays' isn't a thing!
Verbs like aimer and détester always use the definite article.
E.g. Il aime la nourriture. Je déteste le brocoli.