Is there a good free/paid webmail service that can be used with a custom domain and has good security? As far as I know, GMail can only be used with a custom domain if you convert your account to Google Apps, which prevents you from using some Google services.
I use Fastmail.fm for my email hosting (using my own domain), and recommend them. They don't have two-factor authentication, but do have others, such as one-time passwords. They have a lot of features over and above GMail too (such as aliases and personalities), and much prefer them to GMail.
I'm not sure what services you are referring to, and I recall that being the case in the past, but I can't think of any Google product that doesn't work with Google Apps accounts.
You can create a new account with your new domain. Switching among google accounts is pretty easy now, it's just a dropdown in the upper right. You can load Google Docs or email or whatever just by picking an account from a menu.
