How deep should drain pipe be buried under an asphalt driveway?
In my opinion you should be good as it was for depth, moving to Sch 40 is a good upgrade, that is what the company I worked for always used under slabs and especially driveways. 

The main thing is that there are no low areas for water to collect and freeze. Depending on your area, it would be a hard freeze to get ground to freeze at the 12" depth, there would be exceptions as there is always is, but in a hard freeze there should be nothing running off the roof to freeze in the drain.

Although not needed, to increase the fall would be good for heavy rains, but for freezing, the benefit would be marginal.