I have used vallejo primers many times. I have used black, grey and white and also other colored primers. It never caused problems but from my experience I can say they do rub off easier than spray can primer. That might have happened because I tend to thin (to avoid dry-tip) them and they are not meant to. If you do thin them, use airbrush thinner and no flow improver (this slows the drying time, and that's not what you want with a primer). I would always let the primer layer dry 24 hours or at least overnight before you go to town on them. I don't know if it's just me but it seems like the primer rubs off easiest on colored plastic like the plastic used for warhammer underworlds warbands.
I would also advise using the grey primer instead of the white if you want to go light. The grey is really light but is much easier to use then the white primer which has super large pigments. It will dry your tip often, especially when you don't thin it.
Just my 2 cents. Hope this helps a bit. I found airbrushing pretty daunting in the beginning. It would have helped if I knew this back then.
Edit: to answer your second question; when I do zenithal priming I use a black primer. Let it dry. Then use (non-primer)cold grey (it's a better tone between black and white then grey primer which is almost white) and (non-primer)white on top.