Hi Rick,
That’s some good questions …
1) The little blue squares on the Mona’s drums skins are called “Moon Gel” and are basically just little damper pads made out of some sort of squishy jelly gel. You put them on the drums to take away some of the overtones of when you hit a drum, making them sound less “boomy” and more dry. In most cases that’s a preferred sound for drums.
Most drummers use some sort of dampening. Ringo for example often put his wallet, John’s Harmonica case or a cigarette pack on his snare drum or tea towels over his toms. Some people tape down napkins on their drum skins to “flatten” out the sound.
2) The boxes facing the musicians on stage are called monitors or “floor wedges”. As you said, they are there to allow the musicians on stage to hear whatever they need to hear to perform. If you’re on a big stage and the PA (the big speakers) is facing the audience you actually hear very little nuance of everything that’s going on on stage. Especially if you are playing loud music, with a loud drummer and have all the amps on stage. You will hardly hear yourself sing for example which can cause you to be out of tune. That’s why good stage sound and a great sound tech can make or break a show.
Normally each musician has their own speaker facing themselves where the sound guy will “feed” them whatever they need to hear most in order to perform best. Normally that’s mainly your own instrument + voice and the right balance of the other instruments being played on stage.
3) That could be one of two things. Especially in loud, crammed venues some musicians simply put in protective earbuds to look after their ears on stage. If you are standing an inch from the drummer hitting his cymbals with full force every night, you’ll soon damage your hearing, so over time it’s important to protect your ears.
However, most of the time it’s an “in ear monitor” system which does more than simply filter the sound. It’s used instead (or sometimes on top) of monitor speakers. So instead of having a monitor speaker facing you, you get your stage sound through earbuds. So you can for example hear your own voice and guitar playing amplified in the ear phones while it simultaneously filters out the loud frequencies of the drums or bass you might be standing in front of.
It’s a great way to reduce the overall volume on stage and can give you a much more precise mix. Some people (me included) don’t like that this can sometimes take away from the atmosphere as it blocks out some of the crowd noise. So I generally prefer floor wedges over in-ear monitoring.