OK, I can't duplicate your issue with the Group field
What value do you have your field set to? And I presume it is hidden?
Can you double check the Access on that field?
BTW, I don't typically recommend using that, unless you have a very specific need to be able to set the group membership on a per-submission basis, rather than just using the static 'default group' setting on the plugin.
The only wrinkle there is you need it if you do the "syncronize users" thing, i.e. for populating your Fabrik table from the #__users table on first load. But you can get rid of it after you've done that, if you don't need it.
The reason being, it kind of adds a potential vector for hacking, by taking group membership from submitted data (although we do have code to hopefully prevent people from making themselves admins, or elevating their group membership when editing an existing form). So although we think it's safe, there is always the possibility we haven't coded defensively enough.
So I usually recommend leaving that blank, and using the static default instead.
Anyway ... looks like I'll have to debug your site, as I can't replicate this. Catch me on Skype ('cheesegrits'), and we'll have have look.
-- hugh