When starting my car my door light on my gauge is slightly luminating. Also when rolling up my windows and it reaches to the point where it is closed and I'm still holding the button the door light comes on. I just had an alpine alarm system installed, could this be the problem.

The alarm could be the problem, if they tapped off the door switches improperly. A diode may be needed to isolate the alarm from the door switch circuit, to prevent backfeeding which appears to be what you are experiencing.
I would take it back to whoever installed the alarm and have them fix it.

thanks mulder