Is this adapter compatible with Windows 7 and Windows 8?

Yes, it is compatible with Windows 7 and Windows 8 USB equipped laptop and desktop computers.

What resolutions are supported by the USB 3.0 to HDMI External Video Card Adapter?

The adapter supports resolutions up to 1920x1200 for PC and 1080p for HDTV.

Yes, it's backward compatible to USB 2.0

double click image to zoom in

StarTech.com USB32HDEH USB 3.0 to HDMI External Video Card Adapter - 1 Port USB Hub - 1080p - External Graphics Card for Laptops - USB Video Card

Connect an HDMI-equipped display through USB 3.0 while keeping the USB 3.0 port available
Compatible with Windows 7 & Windows 8 USB equipped laptop and desktop computers and HDMI equipped televisions and projectors
USB Video / Multi Monitor Adapter / USB Video Card / USB to Video Adapter Converter / USB to HDMI / USB 3.0 HDMI External Video Card
HD playback
Dual or Triple monitor setup for home office or BOYD (Bring your own Device) work stations