It is not a must to adjust action after leveling. However, the nut action is affected by fret wear, so if you were trying to get everything perfect, you would often bring the nut slots down a bit with the level. I would level, dress, and then check the nut action to make sure it is still acceptable. My guess is that it won’t be enough off to bother you.
You can use a triangle file for shaping the crowns. However, I find a crowning file is much quicker and easier to get consistent results. As I mentioned before, if you have a flat fretboard, a file or a flat sanding block will work for leveling.