My Company wants me to install a router they supplied and connect threw my DSL router.
1. Will this slow down my response time on the computer?
2. Will the company now be able to view if they wanted to all of my computers within the house as to what site they are viewing?
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.
The computer that is hooked directly to my router is own desktop...I have other folks in the house who have their own laptop using the same router wireless. So again do you have any suggestions as to how to prevent Big Brother from viewing what other members of the house hold are viewing?
Not really any way around it. Anything that passes through that router can be tracked. Most likely they're looking at active connections and how much data is coming from you to their network.
The only other solution would be to have a software VPN client installed on your desktop computer which would connect only your computer to their network. You would have to ask them if they had other ways and address your concerns with their IT department.