How can I be sure that my car has a burnt valve and if so, what does it cost to repair?

A compression test tells you the compression is low. A cylinder leakage test tells you why. A similar but different hose is connected to the cylinder through a special gauge, then compressed air is pumped in under low pressure. The gauge tells the percent of leakage and you can look and listen in four places to tell if it's a burned exhaust valve, intake valve, worn piston rings, or a leaking head gasket. Burned valves are not very common any more with the better alloys compared to the '70s.

If you like the car and can be without it for many weeks, you might look for a nearby community college with an Automotive program. We were always looking for live work to give the kids real-world experience. They are well-supervised and very conscientious. They will only take work in that fits with what they're currently studying so you could have to wait many months, but the cost would be very low.