1 - Yes, the router will do fine. Personally I would go for a cheaper
Buffalo WHR-G330N so. It's a proven entry level router that is equivalent to the Belkin unit in specification. It's also wall mountable, which has always been a key criteria for me when buying routers.
2 - No, the Belkin software is optional. However, the software is designed to streamline the setup process for inexperienced users.
3 - Yes, all you should have to do is plug it in, turn it on and enter the SSID (network name) & wireless key/password printed on the underside of the unit into Windows (click the "Start button", then "Connect to" and pick the new network from the list).
4 - Assuming you are running a recent version of Kaspersky (2011 or newer) you should not have any problems at all. Older versions may give you pop ups about asking permission for applications to connect to the new network/Internet connection (by the way, upgrade to the 2012 version is free if you have a valid subscription key).