The DOCK icon counter is a bit of a known issue that seems to happen to some people.
It can be solved by deleting a .plist
This is the com.apple.imessages.bag.plist in the ~/Library/Preferences.
Access this folder in the Finder > Go Menu > Go to Folder and typing that location in the Dialogue box.
Drag the .plist to the Trash then restart Messages.
AIM Messages are ot part of the Syncing process that involves the iMessages account only.
In Messages > Preferences > Accounts > the AIM account > Server Settings is it using SSL to Login ?
The Buddy List would normally close (Contract) if you were being dropped by the server (The SSL one does tend to do this more)
Try it without the SSL (You have to Log Out first)
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.2)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad
"Limit the Logs to the Bits above Binary Images." No, Seriously