I am the executrix of my mom's estate. A family member had her take out a 2nd mortgage on the house and after her death stopped paying. I tried to pay it but the amount of payment went up over 1,000 per month. I have had a lawyer working on the case but he hasn't been very helpful.

If the house is taken by foreclosure, can it still go on my credit report even though my name is not on the mortgage and the bank refuses to negotiate with me?

No, your mother's debts do not affect you, even as executrix. Is there any proof that the proceeds of the loan when to this family member and that the family member agree to pay the loan? If so, the estate can file suit against the family member for payment.

Is the house upside down (more is owed then the market value)? If so then I would let it go to foreclosure. Did mom have any other assets?

No, that is your mother's debt and the house is in her name.