There are 3 separate issues here.
1. Why you can't do Internet sharing.
The modem/your ISP can only handle one IP address. This is presently your PC so no other devices can get access.
Your Linksys router has to be between your modem and the rest of the network with DHCP enabled (and your log-on information). It will then appear as a single PC to the rest of the world and you can hang as many PCs as you want off the back of it. It hasn't got a firewall, but you can limit access to your laptop MAC addresses.
2. How to connect the router to the USB modem.
There are various USB to ethernet convertors ( see
here, but ensuring it works and there are no problems is problematical. personally, I'd advise getting a new UTP/ethernet modem. Depends if you want to take the chance and spend around £70 instead of £25. e.g.
Modem
I'd speak to the Lanshop and see what they say.
3. Why your present system drops the laptop link.
Windows XP authentication has caused problems with the WRT54G, with several reports of dropped links. Try disabling the authentication and see if it stops happening. If so, try altering the WEP settings and/or see if you can download a firmware upgrade.