By asking someone to sign a contract, you are telling them that you are going to sue them if things go poorly.
Not necessarily. I think she is thinking about this from a financial self preservation mindset, which I can understand. What she may not realize is that it does get muddy if say they get married. No state that I am aware of allows one spouse to evict the other, and living together without getting married is also a sticky wicket that way because it is not strictly and landlord/tenant interaction (so I'm not sure it would hold up in court.)
But that really is HER problem, not OP's. It's a trust question from her standpoint. Does she trust him enough that he won't screw her over financially?
What my friends who are getting married this weekend did, was she kept her house (in her sole name, not community property as it is wholly owned prior to marriage), she moved into HIS house, she signed a prenup about other assets and what happens if things don't work out, and she rented out her house to a tenant. He has a much greater net worth than her, and they both have grown children they wish to leave things to, and this makes sense.
My advice in this case would be not to move into her place, but I do understand that this creates undue costs that are avoided if y'all cohabitate (and I understand OP may not own his own place), but if you go half on rent/bills then the cost is what it is. A roof over your head is going to cost something anywhere you live. Sounds to me like she may have had a moocher somewhere in her past and she is now trying to protect herself.
My father always said that the time to draw up a partnership agreement or contract is when everyone is getting along, because everyone at that time expects things to go smoothly. The agreement outlines what happens if things sour, which no one expects going in, but the agreement needs to reflect what happens in the event that it does.
OP would you willingly sign a prenup at some point (assuming you two marry) stating that the house is her sole property independent of you no matter what (including divorce)? If so tell her that.
If you are unwilling to sign such a document, then she is correct to try and protect herself here.