Hmm.. we've never seen the double facebook button, anywhere. We use Chrome most of the time too, but will keep an eye out for possible explanations. Thanks for pointing it out, but I have no suggestions on what it could be..
As for the Apps directory, you don't need to submit it at all. We don't submit any of ours. It's beneficial to submit them if you want them searchable by users on Facebook, which is good for games or things like that. It can help you with getting users to your site, a little bit, maybe, but we haven't seen any big advantages to this unless your application name is something people search for (either well known, or good keywords).
All in all, submitting your app to the director isn't required, and you can always decide to do it later. I can't remember if there's still a fee for submitting it, but there was at one point if you had less than a certain number of users.
Hope this helps, and good luck!