Uwe,
You can load the Javascript library multiple times, but they absolutely cannot have different application IDs. This will confuse Facebook and cause the issues you're having. They will be 'random' because sometimes, for some functions, the right AppID will be used, and other times, the other 'incorrect' ID will be used for whatever your trying to do (Like, Login, etc). On your page, I see the following 2 appIds:
192463310792363 - Used by JFBConnect
---- Which goes to this App Page:
www.facebook.com/apps/application.php?id=192463310792363)
e8a3aa855c603c825e217d34d8685fc7 - Used by the "FB JSDK Kit"
---- This app page redirects to FB.com for some reason:
www.facebook.com/apps/application.php?id...3c825e217d34d8685fc7
Is there a reason you need 2 different applications for 1 site? Generally, you should try to consolidate these so that all your information, user's Like'ing, Logging in, and general promotion that can be done through applications is all contained in one place. JFBConnect doesn't care which of the above you settle on, but you should choose only one.
Hope this helps explain the problem, and a possible solution.