Let me format your post as a list of questions:
#1. What leads us to believe that there is a black hole in the center of the milky way?
Studies directed at the center of the galaxy have shown interesting behavior in the movement of stars close to the core. They then realized the cause of the orbital movement of the stars must have been due to a BH. The specific observations made can be readily found in the wiki:
http://en.wikipedia.org/wiki/Sagittarius_A*
#2. If there is one, why is the center of galaxies always depicted as being a bright sphere?
I think you're referring to galactic haloes or bulges? The density of stars as we get closer to the center of the galaxy is larger and so the intensity should increase. In fact in order to study sag A you would have to find a wavelength that can penetrate the dust gas and stars closer to the center. I believe they used radio frequencies for this.
#3. Is it a rotating black hole, and how would've it formed?
This is one of the biggest puzzles in astrophysics and in fact I remember doing a paper about it in high school. It's the chicken and egg theory, which I'm sure you know "Which came first, the chicken or the egg?".
Well in this case the question becomes "Which came first, supermassive black holes, or galaxies".
There are a few theories regarding this. One of them obviously involves a black hole of a normal size (> 8 solar masses but < 100,000 solar masses) that has accreted material slowly over time. Another is the collapse of a giant cloud of gas into a star of super massive size which is so unstable that it becomes a SMBH sans the normal nucleosynthesis and supernova processes. There's also the theory of black holes that formed shortly after the big bang due to the high density of matter in that period.
For each of these cases the galaxy came second, meaning that once the SMBH had formed, it had accreted material around it that eventually formed into a galaxy.
The other theory obviously is the formation of a galaxy as one would think, but with a high density of stars near the core that cause a massive collision and collapse into an SMBH.
