Brake pedal pulsating indicates a warped brake rotor. It's not going to get better or go away. You can try to have them turned but if it's an old rotor, it may fall out of specs. New rotors aren't that expensive though. They are not that hard to replace. Might as well replace the pads while you're in there and call it a brake job. Lol.
Does it have drums or disc on the rear?
The moaning sound may very well be the booster. Check for an air leak at the vacuum connections first. Generally when a booster goes bad the brake pedal will feel like you are standing on a rock. If it's still a normal feeling pedal, go after the pulsating problem first and see if anything changes.