1. Yes, as a matter of fact, it is mandatory to connect the Slingbox to a router in order to get Internet access.
2. The router should have at least one ethernet port. If it is already taken you will need to get a different router with available ethernet ports.
3. The answers are the same if you get the Slingbox PRO HD.
4. The Slingbox PRO HD need a higher connection speed since you will stream a higher quality video signal. The following link will allow you to get the Tech Specs for the Slingbox PRO HD.
You can also check the following link in order to get information about the available resolution options.
5. Not sure if this is what you mean, but if the screen is bigger, the video signal will also be bigger.