A: In theory, if there is no property, alimony, debt, children, etc. to argue over, one spouse would move out and you'd file for divorce after one year of living in separate homes.
In practice, many spouses claim they have no property and this is never the case. Almost everyone has at least a bank account their paychecks go into, transportation like a car (or even a bicycle), etc.. Even the clothes on your back are technically marital property.
That being said, many spouses agree that they will each walk away from the divorce with whatever property is in their name/possession and sign a very simple separation agreement when there isn't a lot of property to fight over.
A: You are both entitled to live in the marital home, regardless of whose name is on the mortgage, until a court orders otherwise. Emotional abuse is not an actionable legal claim. If there is "marital misconduct", such as violence (or cheating) though, especially if there are medical records, police reports, witnesses, photographs of injuries, etc. you can ask the court to evict your spouse through what's called a "divorce from bed and board".
A: I typically attach a copy of the separation agreement at the end of the initial divorce filings as an FYI. However, most separation agreements are private contracts between the spouses that do not involve the court directly. That is, unless your agreement requires "incorporation" or the spouses agree to incorporation. Incorporation is where you convert the agreement to a court order.