Check these two threads: Permissions & also icon bouncing in dock
May I recommend you try using the Standalone Java installer b/c sometimes updates over the internet cause some side-fx.

What type of firewall do you use. Have you configured it to allow LW & Java full access? http://www.limewire.com/english/content/firewalls.shtml

You should delete the icon from the dock when you uninstall an app.

You use dsl, may I ask which modem/router brand & model number you use? Many dsl modems use NAT firewalls.