The Rocketdyne Tripropellant rocket had great specific impulse, one of the best. But-- there are many reasons it never caught on: one of the byproducts was FOOF, along with other things like hydrofluoric acid.
If you want to actually get momentum out of a rocket, the reaction products are going to touch the combustion chamber walls and the nozzle. While film cooling can help with minimizing heat transfer from the hot stuff, I doubt it’s enough to keep this stuff from eating your engine from the inside.
They actually did build a test article and ran the engine a few times, enough to gather the data but it indeed ate the engine, and the concrete and the rocks and coated it all with explosive powder.
They did imagine coating the proposed launch complex with quartz but it quickly became obvious it was going to be way too expensive to actually build.
Yes, many. It's widely used on solid rocket motors, but it's considered a bad idea on liquid fuels because regenerative cooling is usually lighter, more efficient and easier to design and build. It's hard to get a rocket nozzle to ablate in such an even and consistent way that you wouldn't have to provide much larger safety margins than you'd really want to in a rocket.
Arguably, the solid rocket motor is in this vein. While I've never seen a design that consumes the outer shell, the inner material is designed to burn as completely as possible, and the chemistry and physical composition is even designed to cause the burn to happen in a proper combustion-chamber shape.
