#1
When I pick single notes, I notice that my index finger will sometimes brush the string higher than the string I'm picking and that sometimes my thumb will brush the string lower than the string I'm picking.

Is this normal or is it something I should work on not doing?

Thanks...
#2
If I understood you correctly, it's perfectly fine. It's a good way to mute the strings you don't want to ring out.
#3
While Sashaafm is right and you can use this to mute strings, you might also inadvertently cause those strings you brush to ring out. You might want to try holding the pick in such a way that there's more of the pick not covered by your fingers and so therefore more surface area to strike the strings with
#4
To make it more clear, when I down pick, sometimes I notice that I'll lay the pick down a little bit and my finger nail of my index finger will rub the string higer than the one I'm picking.