Gary,
Very, very sorry for the delays. You're obviously correct that this post fell through, and while not common, it does happen. If it helps, it's not you, it's us
Now, let's get you fixed up. For the name field, you won't see that in the drop-down. The predefined field of 'name' within Community Builder actually references the main Joomla user's name (in the User Manager). This field is automatically filled in by JFBConnect when a user registers with the users full name (First + Last), and we don't really give users any say about that. If you have separated out First and Last name fields, they should show up in your list, and you can choose to map those however you want.
Right now, for the mapping fields, we show any fields in CB that you could map into (mainly text type or specific 'predefined' ones). We do however show fields whether they are published or not, so even if you're not using the first, middle, last name fields, they'll show up. If you're not using them, just leave the mapping to "None"
As for the avatar, if you don't want it imported, we have an easy option to disable that ability so you don't have to worry about it. Alternatively, you can set the "Always import profile" setting to No. For a site with reservations for things, we'd recommend that setting, so that it only imports the user's profile (and avatar, if you want) once.. and then, if they update bits of their profile, it won't be overwritten later on.
In all, from the above questions, it doesn't seem like there's anything to 'fix', we just need to clarify things. However, if I'm mistaken and you're actually running into issues, just let us know. We'll prioritize anything from you going forward just to make sure we get you going.
Hope this all helps, but let us know if you need anything.
Good luck,
Alex