Foreclosure times vary by State and the laws.
The bank does the foreclosure action and at the end of the action you are no longer in the house. There is no discussion.
If you go to foreclosure, how are you going to get another house? Your name only? who will have the credit? Can you get a fixed rate Mortgage (fannie mae etc.) ?
Google "foreclsure in MI" and see what you learn.
It sounds like you have a variable rate mortgage.. and with a mortgage company and not a bank. If you read your contract, missed payments mean an increase in interest and it can be high (as you have found out).
No matter what you decide this money is owed. If you sell the house you are in, the money owed will come out of the sale (taxes, mortgage, $10,000 loan etc.). If the house is not worth what you owe, the bank may put a lien on anything else you own and you may have to sell those things or have them seized, to satisfy the loan. If you buy another house, this could mean the new house could have a lien placed on if from the old house (tho in NY the old lien would likely mean you would not ahve credit for a new house). If you are married, they may be able to come after you as the spouse.
I don't know the particulars (numbers) and I don't know MI law but I do know that mortgage companies have had some issues of late.
LEARN YOUR RIGHTS. Get some legal knowledge sooner rather than later.