I wouldn't worry about connecting to the internet, I have my main DAW connected to the internet all the time even when I am recordging, it's all myth's and fairies about some great big nasty virus is going to eat your computer. Just don't accept that offer of marriage from that russian saying she is a princess.
Welllll, that depends. A properly protected computer isn't at much risk from a random attack (firewall, virus protection, etc.) But there are still good reasons for disconnecting from the Internet. Some background services routinely reach out to the Internet, which can sometimes spike the processor or disk access. Plus, all that anti-virus crap is horrible about interrupting the system...so you'll want to disable it, and that means you probably want to disable the Internet, as well, at least while you're working.
I admit I leave the Internet on sometimes, too, but there have been times I've regretted it (catching a spike right in the middle of recording the perfect take, for example). So, I prefer to minimize the risk whenever possible. Naturally, YMMV.