The full process of forclosure will take about 90 days, so, if you pay the $2,000 or even $1,000 you will delay the process by 30 days....the mortgage company does not really want the house, they are not in the business of selling forclosures, that is not their bread and butter. I'd do as Out advised...and then no matter what, I would pay them at least a 1,000, toot sweet. It shows intent.
If you default, you will have a big problem fianacing another place. Even owners of rental property do credit and background checks and there is a certain credit score that they have established to approve your moving into their complex. And all adults who will be renting must comply to the criteria.
To buy a home, you will not only have to prove that your income can support the loan but that you are credit worthy.
Have you checked out filing bankruptcy, in most states you can keep your home and one car.