If you only ever use the weight of the blade you'll hone very slowly; this is probably the right technique when you're starting, just be aware that it's gonna be slow so don't get frustrated; going slow is better than honing your razor into a steel toothpick.

The gritty feeling's a problem, it shouldn't be gritty at all. The 600 grit paper is fine for lapping, that's what I used, and if you read the directions that came with your stone, that's what norton recommends. But it sounds like you do have grit in the stone, and this will keep your blade from getting sharp, so you need to get it out. Brush it more with the scotchbrite pad *under* *running* *water* and see if that helps.

It's not required that the bevel be the same on both sides. It's not unusual for a razor to be a bit warped, but even so it's not unusual for one side to have a larger bevel than the other side, or for the bevel to be larger on one end and smaller on the other (so the edge is running kind of diagonally). Very few of my vintage razors have even bevels. It's not desirable, but it doesn't seem to be a big deal either. Still, I try to make sure that I'm not the one causing the uneven bevel, and I try to hone in such a way as to improve any unevenness in my vintage razors.