什么样=which kind of. It is quite right that you perceive 什么 as the marker to put up with a question(to query), for example, 什么是你最爱吃的水果（What is your favorite fruit?) . But sometimes it could be general word,這句話沒有什麼道理。(means, "this sentence do not make sense, is nonsense"). Basically, 什么+noun=which( kind of) + noun. So, " aspect of his home is what he picky about "-----it suddenly up to my mind that the point tend to be that you omit "对"（to, with regard to), if put that sentence like 他（对他住什么样的地方, with regard to which kind of place he lives in）很挑剔 ， it seems to be easier to understand. May this help u^-^

他出国玩的时候，他对住的地方（环境）很挑剔。

這裡的 什麼 不是疑問詞 而是 不定的形容詞 用以 形容 其後的名詞 樣 When he goes out to play (travel), he is picky about the place where he lives. 這句話沒有什麼道理。 這個人沒有什麼腦袋。(腦袋=智慧) 為了這點小事，弄得人家雞飛狗跳，也不算有什麼作為。

他出国玩儿的时候，(他)对他住什么样的地方很挑剔. When he travells abroad, he is very picky about the places to stay.