use a proper facebook messenger, it's better plus has more features.
use this http://www.facebook.com/apps/application.php?id=102332746051&sk=info
you can be invisible while still seeing who is online. And be online to some friends, while appearing offline to others.
Also has a cool away/busy function.
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You can't become "Invisible" on Facebook Chat. Instead you can appear "Offline" though still be on Facebook, just others cannot see that you are online.
Yes you can. Lets think of it this way. Say you were chating to a person on Facebook and you logged off without clearing the chat or exiting the chat.The next time you log on if the person is online or offline your chat converaation will still be there. Hope this helped! :)
When you send a message on Facebook, your message is private between you and the recipients. It will not appear on the recipient's wall.
They actually can, but it appears from a person with no link (i.e. name is in black) and no photo
Yes. You can use Messenger using a mobile phone number. Just tap the "Not on Facebook?" button.