This could be several problems at once (the shutting down thing being one problem, and the internet being another) or it could be a single virus causing both issues.
As Scott said, try doing a forced shutdown. If that works, you could try starting the laptop in safe mode to see if it's any better.
As for the internet problem, how do you connect to the internet? i.e. is it via a router? If so, try logging on to the router and see if 1) you can access it normally, and 2) see if the router is connecting properly to your ISP.
You could also try opening a command prompt and pinging a well-known coprorate website such as "www.bbc.co.uk".