The last 2 japanese sentences are not very accurate; I had the room cleaned for my mother. 母のためにそうじをした I had him clean the room for my mother. 母のために彼にそうじをさせた

I cleaned the room •(私は)部屋をそうじしました。 I had the room cleaned •部屋をそうじしてもらいました。 (*部屋をそうじさせました = I made someone clean the room for me. ) **It's the basic translation. My mother cleaned the room for me •母に部屋をそうじしてもらいました。(or 母は私のために部屋をそうじしてくれました。) I cleaned the room for my mother •母のために部屋をそうじしてあげました or そうじしました。 I had the room cleaned for my mother •私は母のために部屋を(someoneに)そうじしてもらいました。 I had him clean the room for my mother •私は母のために彼に部屋をそうじしてもらいました。 *I wrote the most basic translations. I think there are other translations depending on the situation, for example, the subtitles in the movie would be different. I hope this was helpful.

I apologize that the latter part of the comment is unfinished. I agree with your argument. 部屋を掃除させました sounds natural.

あげるhere should be written in hiragana, not kanji. When it does, it's 上げる. 挙げる is incorrect. I had the room cleaned for my mother 母に部屋をそうじさせてあげました incorrect. This Japanese sentence mean"you had your mom to clean your room". So who cleaned the room is your mom. 母のために部屋を掃除させました。or 母の部屋を掃除させました。 could be better. I had him clean the room for my mother 彼が母に部屋をそうじさせて上げました incorrect. This means "He had his mother clean the room" よろしくお願いします I have just realised that the English in the second sentence is ambiguous. I think "Someone cleaned the room for me" is probably better. "I had the room cleaned" in English means more like, "I arranged to have the cleaning done", so that would be "部屋をそうじさせました’ ?

I'll let a native comment on the correctness (but they look correct to me). I have just one comment though. In my experience, when the word あげる means "give" or used to denote an action that you perform for someone, then it usually doesn't use the kanji and when it does you would use 挙げる.