Yes, you should keep to the 'conventional' order of things first use the windows add remove programs, the uninstall tool is a belt and braces approach as there are a multitude of posts in the forums where remnants are left after a conventional add remove uninstall.
I can't say for certain, I have never used Norton or consequently the uninstall tool, but I believe it is selective as opposed to everything.
If completely removing NIS you will have to consider a firewall as avast doesn't include a firewall.