Sorry for all the troubles. Going to put a few changes to do in this post, but still may run into issues. We're working on a new Invite module for the next release, as this one uses an older method from Facebook which as you can see, has some problems.
In plugins/system/jat3/base-themes/default/css/layout.css :
* At line 107, you'll see .ja-box. Change overflow:hidden to overflow:visible.
* At line 18, you'll see .column. Change overflow:hidden to overflow:visible.
In the JFBCInvite module, set the rows to 3.
One final thing. The JFBCInvite modules' template file seems to have been changed, from what I can see. There's an overflow:visible in there that should actually be overflow:hidden. If you did that, can you change it back? If you didn't, can you simply re-install the JFBCInvite module to reset it to how it should be?
Again, sorry for the troubles, but it will be easier to diagnose things now. Once we get it working one way, you can try messing with the columns and rows a little to find other settings that may work, but I think the above should do it.
Thanks,
Alex