1) For those schools that are "Interview by Invitation ONLY", is it necessary or will it significantly improve your chances of admittance to visit these schools prior to or during the application process (September/October)?
It is never necessary, but it always helps. I do NOT think it "significantly" improves your chances.

2) For those schools that are NOT "Interview by Invitation ONLY", is it necessary or will it significantly improve your chances of admittance to visit these schools prior to or during the application process (September/October)?
Same answer!

It's obvious that by visiting schools you have a better feeling of whether the next two years of your life would be best spent at a particular school, but FOR ADMITTANCE PURPOSES ONLY, how do schools feel about those students who do not voluntarily visit schools prior to or during the app process?
It is always nice to see that you visited, did your homework, are legitimately interested in a school and that you are genuine in wanting to apply and potentially go there. It definitely can make an impact. It also improves the quality of your application to have real insight into a school and to be able to convey that in your essay. So it does help for admittance purposes - both in that it shows "your love", and it improves your application.

Correct me if I'm wrong, but I would guess it can't hurt to show the school some love by displaying interest through a voluntary visit, but is IT that big of an advantage in terms of an acceptance?

Not a huge advantage. But in this competitve game, everything helps.