With regard to the clarinet and saxophone, what are the advantages and disadvantages of a more open mouthpiece? I seem to get a very good sound playing with a fairly closed mouthpiece, and the volume is fine.
I can speak more about saxophone that clarinet since that is what I play. I wouldn't say one is better than the other; just a matter of what sound you are looking for. I play a fairly large chamber Tenor mouthpiece (Otto Link 7) since it gets me a huge sound which is what I want for tenor. Being able to put more air in the horn as a result makes the sound fuller and rounder. In regard to clarinet, I heard Eddie Daniels say that when he was recording his album "The Five Seasons" he needed to switch to a mouthpiece with a closer facing which didn't make as much sound in order to conquer Vivaldi's Four Seasons. See here for more detail.
There is a lot of difference and preference player to player. I've found on saxophone, a more open mouthpiece facing allows me great control over my tone, pitch and volume than more closed facing mouthpieces. I used to use a great metal selmer soloist faced to a D* for classical (one might consider quite open in the classical world). I had a friend who could barely make a sound on it, but when they played their selmer ss soloist c it would bring you to tears. I never got more than a whistly squeak out of it.
The general rule for mouthpieces is find the piece that works for you and NEVER LET IT GO.