No it won't. The wavelength of UV light excites phospors in the paint. Strobe lights do not work on these wavelengths so you will just see the same colour paint as you would under normal lighting.
It would, I agree. But in all my years of working in the entertainment industry, I have only ever seen Xenon flash tubes producing white light in strobes. I am unaware of a UV strobe's existence.
It will depend on the length of emission from the buld. Xenon bulbs can emmit from 250nm to about 800nm, peaking in the vis. I would expect most to have filters for the higher energy UV (250-400 ish) but you may sill get significant UV emission. Whether it will work with the paint???? Not sure.