With JFBConnect, you can enable an auto registration (like we have on this site) by setting the "New User Creation" to "Facebook User Only". That will automatically pick the user's name and just log them right in.. no forms.
As for auto-registering new users that go through the 'normal' Joomla process, generally, I wouldn't recommend it. You'll get a ton of spam users eventually, and once it's found that your site is easy to register on (and heaven forbid, actually post any content too), you'll get marked as a spammer's paradise. Once this happens, even if you rein things in so that it's not easy to register, you're site will already be on some lists for spammers, and it will continue for a long, long time.
That's one of the advantages with JFBConnect, and why approval isn't necessarily required. It's tied to a Facebook account and you can get their 'real' email address. We've never heard of spammers using their FB accounts, so it's been much more safe/secure in general.
As for using the AJAX register component, no, it shouldn't cause problems. The only problems that arise are from plugins that redirect the user. That's simply because they fire in the middle of JFBConnect's registration process. If that happens, the user is redirected and the rest of our registration code doesn't execute.. which is bad.
Hope that all helps,
Alex