I think :
- A woman who you rent a room
or A woman that you rent a room
or A woman you rent a room
A woman is the object .... you is the subject .... so if you said :
the woman who/that lives in ....
you cant leav out who or that ..so you can't said : the woman lives .. incorrect ... because the woman is the subject
- the hat jim was too big for him ... or ... the hat which/that jim was wearing was too big for him
or .... the hat jim was wearing was too big for him