I would suggest Debian. Rock solid and it usually has all you need. If you need newer software on it you can allways use backports and there is also a really cool project called distroboc that will let you run pretty much any application in a container.
I gave up on firefox 1 year ago and went to the dark side with Brave. I am really happy with it even tho part of it is closed source.
You should split groceries/electricity and all that 50/50. The house is yours so she should not pay for it unless you actually both agree she should pay a monthly rent to live in your house. That to me would be correct. I am in a different situation as you as I bought the house with my wife and even tho I spent more because I had more money available I did not ask her extra money. Also keep in mind that wife is different than girlfriend. So since wife>girlfriend I honestly thing you should get to an agreement with your girlfirend and she should actually pay something for the rent and it should not be all on you.
I will take a look, it looks interesting. Thank you very much