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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.

Google Apps has aliases and catch-all.

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.

When I used Apps, I just created a new account and used a different browser window (in Incognito) to access Gmail.

What do you use now?

I run my own server. exim4 and Sup[1] on Debian on a $3/month VPS in the Netherlands. Access to the client through SSH.

[1]: http://sup.rubyforge.org/

You can use fastmail.fm with your own domains.