Yes, you can use the 54Mbps wireless adapter with the 300Mbps router and the link will be 54Mbps.
As for the drivers, it should be available on the CD, but your best bet is to download the latest version as the model printed on the device that goes with your operating system. You can do that with the laptop and move it with a storage device like a flashdisk, or temporarily use a network cable. Go through this page: Belkin : Support F5D7050 Wireless G USB Network Adapter - Drivers
Hopefully your system is supported with the version of the wireless device you have.