Jonathan,
you've asked some simple questions that really have some complex answers. I'll try to answer what I can.
Beating upwind, releasing mainsheet or turn upwind or both? You got it.
Running downwind: first of all you normally want to increase the fullness of your sails downwind. You may already know this but the typical 'running' straight downwind you see monohulls do doesn't apply to catamarans. Our boats are so much more efficient that we can actually get to a point downwind faster by sailing angles and jibing (zig zagging) to get there. Most cats can actually move faster than the wind speed like this. I'll be glad to go into more detail about why this is possible if you ask. The reason you want more fullness / power in your sails downwind is because the angle the sails are driving to is much more forward and less likely to tip the boat sideways. Likewise, there's less leverage between the sails and the hulls in the water. Back to your question: if you encounter a puff when sailing 'downwind' (at an angle to the wind), you want to center the traveler (if you have one) and sheet in the mainsail and/or steer farther downwind. If you are already pointing straight downwind the only way to depower is to bring your sails in tighter as you mentioned.
You are much more vulnerable when reaching....sort of darned if you do, darned if you don't. The best thing to do in an overpowered situation on a reach is release the jib (again if you have one - I don't know what kind of boat you are sailing) to take the pressure off the nose. Then release the main if you still need to reduce power. Usually in a reach the first "alarm" is not the boat flipping sideways but when the leeward bow starts to go submarine. If it goes too far (this is very dependent on boat design) the boat can literally trip on it's nose and "pitch pole" (see image). If you are on a deep reach, steering down with the wind might help. If you are puffed on a reach and you try to turn the boat into the wind, the turning force will only help the boat turn over.
