The information for the Sun and Moon times (rise, set and azimuth) for Mt. Rainier NP for 2013 (January through December) are now available from the Sun & Moon information Web page.
And yes, that's the moon and a planet from the Web cam in the administration building west of the (new 2008) Jackson Visitors on April 7, 2012 at 6:27 am. The Web cam looks southwest down the Nisqually valley to Longmire. Here's the photo 20 minutes later.