Your questions about the vibrato are quite common, especially among younger singers. First I would say that the proper vibrato is completely natural. You should not try to make it. I always describe it as a symptom. It tells a listener what the condition of your voice is. And just as a good doctor treats the cause and not the symptom, we should not worry too much about the vibrato and set our attention on learning the proper functioning of our voice. When that happens and conditions are right the vibrato starts to appear. The actual physics of the vibrato are very complex and not fully understood even by the scientists studying it. But we can say that it comes from the complexity of the vibrating of the vocal folds. Not from moving the jaw or the throat, or from pulsing the air pressure with the diaphragm.
Some people don't have a vibrato because they sing with a slight squeeze in the throat and this impairs the flexibility necessary for the vibrato to exist. So that would be something to check to see if you are doing. Another possible interference is if you are pushing a lot of breath through or against the larynx. This excess air pressure causes the vocal folds to not vibrate freely so they don't recoil, which is a key element in the vibrato. Both of these can be caused by trying too hard. The reason some people have success by faking a vibrato is because it loosens the grip of the throat muscles and can reduce the amount of air pressure allowing some natural variation in the vibration.
As far as breathing, try to to breath slow and complete. Don't just use your abdomen or only your chest. Feel like you breath a little over a large area and not a lot in a small area. Again the vibrato should feel natural, like it happens on its own. Some descriptions would be shimmer, spin, small quiver, vibrancy. Sometimes we mistake the bobbing of our larynx for a vibrato. That is just an unstable larynx. Always strive for a flexible, steady tone production and the proper vibrato will eventually show up.