To start, sorry for the delayed response.
1) The missing check is normal. We didn't put it there because if you're looking at the component, it's installed.. but we can understand the confusion and will likely be adding the check mark in the next release.
2) The "Basic information with name, avatar, etc" is definitely required. My guess is that you already gave permission for your application for this user before you tried JFBConnect, and the Birthday and Hometown are 'new' permissions that JFBC is requiring. Facebook won't ask for permissions you've already granted.
As for the fields not importing, is it just those 2 fields, or are the others (birthday, hometown, etc) not importing as well? Also, can you try creating a simple text field and name it "First Name" and try to import into that field? The answers to these questions will help us diagnose the problem further.
3) That's not good, and could be related to #2. Can you try looking at your User plugins and set JFBConnect to be ordered #2 with the Joomla User plugin ordered #1. The reason I say this is because if you have other User plugins enabled, if any of them try to redirect the user, it will prevent later ordered plugins from firing. If the JFBConnect User plugin doesn't activiate, the profile (and avatar) won't import.
Another thing to check is that the /images/comprofiler directory is writable.
4) There's no way to set the defaults just for a Facebook user. We do set all profile fields to their defaults (from CB) before the user is created, so it should have the defaults from CB. Hope that makes sense..
5) Right now, no. This is something we're looking into adding to the CB plugin. With 3.1 we broke out the integrations into the plugins just so we had this type of flexibility.
6) This is definitely something we'll be adding as a configuration option in one of the next releases. We just started to get this request a bit. Until then, if you'd like to manually make the change, read the post below which should guide you through it pretty well:
www.sourcecoast.com/forums/topic?id=1531&p=1#p6222
If you let the user change it in CB, it shouldn't have any bad effects. We link the users by their numerical database ID's, not their username, so change to your hearts content. Standard disclaimer: test after you make the change.
Hope this helps, and if you have any other questions, don't hesitate to ask. It's what we're here for! If you can't get the profile import stuff going, feel free to send us a Super Admin account by private message, and we can take a look if there's anything obvious wrong.
Thanks!