You won't need to. If you are uninstalling LW, it will only remove the actual program. Start menu > All Programs > LW > Uninstall. Never use the add/remove control panel for LW. Actually, if you reinstall LW, it will remove the previous version at beginning of the installation process.

To uninstall Java, you will need to use the add remove control panel to remove any newer or older versions of Java. Install Java before re-installing LW if you choose to re-install LW.

Also check other options here Fixes for Limewire (click on blue link) as once in a while a change in such settings can also fix connection problems.

When backing up .. do you mean the program or the shared folder? For shared folder, it should be safe on condition you did not make the location either the program folder or the preferences folder.

It might sound strange, but I've heard of some people solving connection issues by installing FrostWire. Then after connecting with it, closing it. Then their LW connected with no problems.