Susan moved into a new city and found a new house to live in.
Is 'in' needed?
(Not a teacher)
Additionally, you could say, "Susan moved to a new city and found a new house." if you wanted to be succinct and remove the preposition. The readers/listeners would assume that you meant a house in which to live.
get rid of 'to live in' entirely. What else was she going to do with it? Pay the rent and sleep in shop doorways? (When speaking a foreign language, more words are just more opportunities for getting something wrong - even though, in this case, it happens to be right. So I'm in favour of succinctness.)