I know for wind instruments and vocals is breathing very important, but how about other instruments? I noticed, that I am breathing different when I play in higher tempo or improvising. I know some actors do squats before performing to get in "mood". Can breathing affect how we express ourselves in music too? Should I even care about it? If yes, how to breath properly and what exercises should I do?
Excluding wind and vocal instruments, I would say precise breathing technique is probably not important for it's own sake. However...
Breathing problems often indicate a more systemic problem with muscular tension, and this can definitely be an issue with any instrumental performance. Especially since, in your case, you seem to be switching technique during more difficult passages (at speed, or improvised). When playing an instrument, you generally need to be as relaxed as possible. For example, if you find yourself breathing from your upper chest and shoulders, you are too tense. A relaxed breath originates from the diaphragm, deep within the body.
I believe different people store tension in different areas. In may case, I often find my shoulders constantly trying to rise, or sometimes my jaw clenching. Even at work (which is computer programming, not music related) I will consciously relax my shoulders (or jaw) several times throughout the day, whenever I become aware of this occurring. For other people, tension can be stored more in the back or neck.
The best thing you can do is to gain an awareness of your body. Learn to recognize when and where you are tense, and make an intentional decision to relax those areas. Take a few deep breaths from your diaphragm to remind your body how it feels. Do this often enough and it will become a habit. As for exercises, I don't think you need to resort to anything fancy. Probably a quick five minutes of muscle stretches and neck rolls will do the trick. At the very least, you want to stretch your arms, shoulders, and neck. Maybe also your back and jaw. Besides loosening up your muscles, finding tense spots, and helping you to breath more deeply, it just plain feels good.
P.S. -- A really awesome way to practice deeper breathing is to just start laughing really hard (coughing and crying hard can have the same effect in terms of activating the diaphragm and clearing the lungs of old air, but laughing is more fun, and more likely to relieve other stress). Of course, I wouldn't recommend doing it around other people, or they might think you're crazy. But maybe while driving somewhere alone in the car...
As a percussionist in a percussion ensemble, I tend to breathe in at the start of a phrase or on the prep beat before a musical transition. I would breathe out after the tension resolves or after I've completed my part of it. Above all, breathing should feel natural. Don't always think that you have to breathe in a certain way for a certain piece, though I find that it is pretty natural to breathe and "flow" with the pace or mood of the music.
Breathing is really just part of a larger category of musicianship, how you can move and react visually and internally to the piece as you're playing or listening to it. Being musical encompasses all aspects of movement during a performance, including breathing. So learning how to control that and just practicing breathing in and out with a song where it feels appropriate and also comfortable can help you segue into understanding how to make a performance visually more compelling, more "groovy". Stuff like swaying to the music or facial expressions, etc. Again, breathing is a nice place to start, so just put on a song or play a simple piece and practice that!