Does the iPad 3rd generation output 1080p video using the HDMI adaptor? I can play 1080p videos I've loaded onto the iPad using the video player on the iPad itself.  Before I invest in the HDMI adaptor, I'd like to know if the HDMI output actually displays 1080p on the external display, and shows the video at full resolution?

I'm worried that either the output will be something lower (1080i, 720p) or that it may be 1080p output, but the video will be resolution reduced to 720p or 480p even though displayed on a 1080p device.
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!