Yes, but make sure the first coat is using paint that has been thoroughly mixed (for this there's no such thing as "too much") and that it's given more than ample curing time before the 2nd coat goes on.
What paint're you using? Jam?
I doubt it, not for the top to a vanity. Jars will be dropped on it.Given the glass will sit on top of an HMR carcass, would 6mm glass be sufficient?10mm is about right, although personally I'd like to go thicker again.
I'd use acid-free Silicone, as used for bonding mirrors. Liquid nails is too strong a bond, with any timber movement either the bond will fail or the glass will crack. Ordinary Silicone will eat away the paint.What about attaching the glass to the carcass - is Liquid Nails OK?