Does 开他的玩笑 mean "he" plays or is played?
'他的' in '开他的玩笑' is not the subject, it is an  'adjectival phrase' 

开玩笑 = to joke 

他的 = (对)他的 = toward/ about him

'对他的' (toward/ about him) is the adjectival phrase for '玩笑' (joke). It describes what kind of joke it is - 'a joke toward/ about him'

'开玩笑' (to joke) is a single verb treated as one word, but in this structure, 玩笑 (joke) is isolated out and treated like a noun, therefore (对)他的 could be applied to it. 

Similarly:

回家 (go home) is a single verb

in '回[溫暖的]家' , '家' is isolated out as a noun and '溫暖的' (warm) is the adjective for '家'(home)