Linear loss obeys Kirchoff's and Ohm's and power laws pretty simply. If you need to lose a certain voltage at a given (max) current, the power lost is P = U * I. For example, 3 V * 0.1 A = 0.3 W. If your LM7812 can run at that overhead (yes), take the voltages and currents (yes) and dissipate the 0.3 W as heat (almost certainly yes) then it'll work.
Caps will depend on how fast the response needs to be. If your load jumps too much too fast and there's no reservoir, you'll get a voltage drop. If it feeds too much back, you'll get a voltage climb. Caps may help and are pretty cheap and IIRC 78xx chips are very stable, but the datasheet will have limits and recommendations.
All the information will be in the data sheet and making some test circuits is easy for these, especially if you have some measuring equipment. Anything below 1A will probably be easy, anything above might be risky, as a rule of thumb. Very small or very large voltage differences will also be tricky or impossible.
Caps will depend on how fast the response needs to be. If your load jumps too much too fast and there's no reservoir, you'll get a voltage drop. If it feeds too much back, you'll get a voltage climb. Caps may help and are pretty cheap and IIRC 78xx chips are very stable, but the datasheet will have limits and recommendations.
All the information will be in the data sheet and making some test circuits is easy for these, especially if you have some measuring equipment. Anything below 1A will probably be easy, anything above might be risky, as a rule of thumb. Very small or very large voltage differences will also be tricky or impossible.