You do not 'live a house' - you 'live IN a house'. 'House' is not the direct object of the sentence, so you have to use the preposition 'in'. "This is the house that I lived in two years ago" is the only correct sentence of the three. You could also omit the 'that' and say "This is the house I lived in two years ago." You cannot omit the 'in'.

Only number three is correct! :)