if I log into neopets in my browser
and I don't ever log out OR clear my cache
and my account says logged in everytime I open neo in that browers
are they taking my ip everytime I go to neo?
No kids. You can login with your proxy and then disable it without consequence. We've been doing this for over a year now without consequence. Login/Logouts are the primary concern. You need your proxy enabled then.
I would suspect that you would also want your proxy enabled while making purchases for NC or Premium as they are likely to be recording those IP's as well since it's a real currency transaction.