I want to install beadboard in a room that has existing baseboard heating units. Is it possible to adjust/space these units away from the wall far enough to fit the beadboard in behind them (without having to do any plumbing work)?

You are asking about full thickness 3/4" lumber... or plywood/ paneling

It's a paneling system. Definitely not 3/4" thick, but not exactly paneling either. Likely 3/8" think in 6 to 9" wide strips.

Copper pies or pex?

Copper