Whether you will owe them for the reimbursement of money laid out depends on the agreement you make with respect to the deed in lieu. After a foreclosure, they will be entitled to a judgment against you for any deficiency, including those costs.
You need to retain counsel, not on the internet.
(1) I may be guessing. Do not act or rely upon this info; (2) We have not established an attorney-client relationship; and (3) If you insist I tell you something upon which you can actually rely: don't eat yellow snow.
Michigan is a recovery state meaning that the lender is allowed to recover the difference between the amount that is owed and the sales price. This would also include any "advances" made by the lender in the form of insurance, taxes, home repair, etc.. Technically you would be responsible for the total difference. However, whether the lender comes after you for the difference remains an open question. The lender as two choices - sue you for the difference or forgive the debt and issue a 1099 to you as debt forgiven. It is unlikely the lender will sue you because they won't be interested in spending more money to get something they probably cannot collect. Unless they are mean and heartless then they can get a judgment that will haunt you for many years. More likely the lender would issue a 1099 and let you have the enjoyment of having a large federal tax obligation for the debt. (Remember- a debt forgiven is taxable to the debtor). However, if the 1099 is issued while the Mortgage Debt Forgiveness Act is still in place (it is at the time of this writing), then you will not be responsible for the taxes on the forgiven debt (provided you provide the IRS with the proper paperwork).