You need to clarify the state.
Yes, unless he/she owns the house.
The options include: stop paying the mortgage and let the bank repossess the house; pay the entire mortgage yourself; divorce the spouse and move out; divorce the spouse and stay, while your spouse moves out; find out why your spouse refuses to pay half of the mortgage and see if some agreement can be reached; seek cheaper housing; go on an extended backpack tour of Europe; enlist in the army. That's about it.
A husband has no legal obligation to support his mother-in-law.
If one spouse moves out the remaining spouse may benefit from having the house all to him/her self, either to enjoy the privacy or to have wild parties on the weekends, one never knows.
In this state, at least, your spouse must file for divorce in the county of residence. All court documents with only a few exceptions (children, sealed incitements) are public record. If a petition for a divorce has been filed, it will be on record.
yes it is alswyas legal as long as you own the house
well...usually the husband moves out (or the wife!!) BEFORE the official divorce so they don't have to live (or sleep) together. but im guessing that they have about a week. the marital judge decides of course.
If you're in the US, no, you don't forfeit property rights merely by moving out of the property.
Darth Vader moves in to your house
Yes, of course (assuming that by "moves in with" you mean "has sex with"). If you haven't divorced your spouse, you are still married, wherever you happen to live, and whatever terms you happen to be on. Indeed, in the Roman Catholic tradition, divorce doesn't exist - you may have your marriage dissolved in the civil courts, but you still remain married in the eyes of God, and it's still adulterous if you take up with someone else. I shall not disguise my contempt for the practice of obtaining an annulment after many years of marriage as a "Catholic divorce".
When papers are first filed, no, not unless the presiding judge were to order it, and he would not likely do so unless one of the parties motioned for it. Once the divorce is final, it would depend on the terms of the divorce agreement and who is going to end up with the house (it may have to be sold and the proceeds divided) or the mortgage. It is likely that if both parties continue to live under the same roof, things could get very acrimonious so from a practical viewpoint it would b best if the partner that is not going to end up living in the house moves out. I've personally known only one couple who continued to occupy the same home after the divorce was final, and they remain friends to this day (about ten years after the fact). Theirs is a very big house.
the one that stays behind.Not the one that moves out