Important mortgage questions

I have a mortgage that was taken out as a joint loan with my partner (we are not married but might as well be). He lost his job about 2 years ago and I believe that when we signed our mortgage paperwork we were told that if either of us changed jobs or lost our jobs it was our responsibility to inform them of this. As we have not been late with any mortgage payments we never told them. But we have missed many opportunities to refinance our mortgage at a far lower interest rate as I have been scared to do so in case they need to verify employment for him. I now need to do some major basement repairs and when I applied for a home equity (for improvement) loan at my local CU they told me that the loan application would have to be a joint one as the house title is in both our names, and that given the fact that he was unemployed at the moment we would almost certainly be turned down for the loan.

My question is twofold- if the mortgage lender finds out that he is unemployed can they alter the terms of our loan? Also, is it possible to have his name removed from the title without involving the mortgage company? I have to wonder what would happen if he died. As a co-signer he would then be taken off the loan and I would be solely responsible, right? But can I voluntarily do this- and if so will they change the terms of the loan?

Any help with this would be much appreciated. We have flooding that needs to be addressed quickly but I cannot do this unless I can borrow off the equity we have built up in our home.