The first and third sound most idiomatic to me, the second clunks a little in my ear.
"I'm trying to find somebody a place to live" works best if "somebody" is specific-- I want to find my friend Ugly Judith a place to live. I would choose this over the other two, probably because I prefer the indirect object, in proper syntax, to a prepositional adverbial phrase.
"I want to find a place to live for..." Now this is the awkward one, and sounds to me most natural if "somebody" is general, as general as possible. I want to find a place to live for everyone who can't afford decent housing. Even better to make "a place" more specific-- "I want to find a decent place to live for everyone who can't afford it."
"I need to find a place for my mother-in-law to live." This is very natural to me, but again, the statments in the abstract sound odd. You don't start out with a need to find someone a place to live, and then get around to the info about who it is.
I don't mean this as a correction, and I know you put the phrases in general form so they could be discussed grammatically, in a definitive way. But the nuances show up in particulars-- at first I didn't think the second example was very idiomatic at all, but when I supplied particulars I realized that the more general or conceptual the place-needing person was, the better it sounded.
Finally, "to live" sounds a little superfluous in the third example. "I want to find a place for my friend" conveys "place to live," and the first example "I want to find my friend a place"-- well, that pretty much does too, but it raises a slight question that "to live" puts to rest. Parts of a sentence that could almost be dispensed with sound better at the end of the sentence.