Does the iPad 3rd generation output 1080p video using the HDMI adaptor?
It works at 1080p. As stated on the Apple web site:

AirPlay video streaming to Apple TV (3rd generation) at up to 1080p and Apple TV (2nd generation) at up to 720p
Video mirroring and video out support: Up to 1080p with Apple Digital AV Adapter or Apple VGA Adapter (adapters sold separately)

Enjoy your new iPad!