1. Is it ok to move into a rental house during 9 days.
2. Would it help if one started the move before the 9 days ?
Although we should not move into a house that we purchased during the nine days, (or the 3 weeks) since it is something that one would say shehechiynu on it. However it is permitted to move into a rental house even during the nine days.
Ohr L’tzion 26-1, Mekor Chaim 552. Halichos Shlomo 14- ftnt. that it is permited to move into a rental apartment, however if there won’t be a monetary loss by moving after the nine days, that is preferred because it isn’t a good siman.