Distribution of marital home

I have two scenarios/questions regarding real estate:

First, if the home was purchased after the date of marriage, titled in both parties names, one party provided a huge lump sum for the down payment, is the party that has been contributing to the mortgage payments be entitled to anything as far as the home goes? FYI, prior to the purchase of the home, the lump sum providing spouse was not the “bread winner” of the family. At most, this party contributed $1000.00 monthly to the family budget for approx. 5 years. The “bread winning” spouse at the time thinks they should be entitled to an equal share regardless of who paid the large lump sum down payment. What does the law say?

Second, if a home was a gift from one party’s parent but titled in both parties names, what party is entitled to the home? If the non-parent party refuses to sign to remove their name from the title, what, if anything, can be done?

In either case the home is marital in nature and subject to equitable distribution. Based on the facts an argument can be made that an equal distribution would not be equitable in this case.