I have two large Autumn maples in my backyard. One has the bark peeling on the trunk with dead wood exposed. Peeling is now moving up limbs. Some leave turning brownish already. Second starting to do that but also has a nasty black fungus(?) coming out. It has been recommended to cut them down by local company. Issue is they provide much shade (40-50 ft tall) and cost of removal. If the trunk/limbs break - major damage to the house could happen. Should I have them removed?

You need to compare what it would cost you to remove it now and what it would cost to repair the house. If you have any doubt, get a second opinion from a certified arborist. You may be able to have just the bad limbs removed. I vote for safety first!