1) Not really. The benefits far outweigh the negatives which are minimal.
2) Depends if they're monitoring you. I'm guessing they're giving you a router to create a VPN tunnel into the company's network. Ideally, yes, they would want to monitor what you're doing while working because you have a hole punched into the company network.
Now, as someone who does this, I can assure you that we as "big brother" do not sit around all day monitoring people. If an issue pops up, then the monitoring will start.
It sounds like you're going to have to hook the router up for work reasons anyways.
If this is your personal computer, do what you want with it unless otherwise stated in your contract. If this is a work computer, use it strictly for work. And last, pay attention to your contract and obligation to work. Are they paying you to do non-work related things while you're working? Only you know, otherwise ask HR.