Go into LimeWire, click on 'help' then choose 'about LimeWire'... that will tell you what versions of LW and Java you're using.

Which company provides your internet service? That's your ISP. It's probably a good idea to call them & find out whether they block or shape bandwidth for peer-to-peer (ie LimeWire). That way, we can rule this out as a cause of your problem. Some satellite providers do limit p2p!

Your modem/router is a device (usually just a flattish box with LEDs) that will plug into your computer & also into your dish/antenna. Try to find the make & model.

Next time you see the error popup, write down exactly what it says or take a screenshot of it.