It sounds like you have a bad neutral safety switch on the transmission. When the reverse part of the neutral safety switch fails, it causes the reverse lights, brake lights, and I believe also some of the gauges to not work. The car can still start with part of the neutral safety switch being bad.

This may not be your problem, but it is my guess. I had the same thing happen to me and that's what my problem was. The oil pressure and speedometer are mechanical gauges that do not use electricity.

It sounds like you have a bad climate control servo (located under the hood on the right [passenger] side of the firewall). It's a very complicated heater valve that uses coolant, electricity, and vacuum to maintain the coolant flow into the heater core.

I don't know the answer to your other question because I'm more used to working on diesel engines.

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1967 W110 Universal Wagon, Euro, Turbo Diesel, Tail Fins, 4 Speed Manual Column Shift, A/C
1980 W116 300SD Turbo Diesel, DB479 Walnut Brown, Sunroof, Highly Optioned, 350,000+ Miles