There are no requirements on your mother issuing a quit claim. She'd be foolish to do so with a loan still in her name. Also depending on the situation there may be other tax and long term care issues that make such a transfer ill-advised. If she is of the medicaid age or intends to continue residing in this house she may wish to talk to someone about this and what she is trying to accomplish (there may be better ways to get the house to you than just outright gifting it).
If she still wants to do that, then she should do the deed at escrow of your refinance.
Whether anybody will finance you given your situation (there's more to it than debt-to-income), we can't tell. You won't know until you start inquiring at lenders.