After 9 years exposed to unfiltered sunlight and radiation on the surface of the moon, the photo of Deke Slayton would have turned completely white.
Speaking of weight:
When they show the astronauts inside the moon base, they walk as if they were on earth. Even though they are inside, the moon gravity is the same as outside: 1/6 of earth's gravity therefore they should be walking the same "clumsy" way as outside.
When Tracy is being interviewed by Johnny Carson, one of the other astronauts on Jamestown comments that "I like Letterman." While Late Show with David Letterman (1993) didn't premiere until 1993, Late Night with David Letterman (1982) ran from 1982 to 1993.
When Ellen is at the bookstore waiting for Pam, you can clearly see Gone Girl by Gillian Flynn on one of the bookshelves. Gone Girl didn't come out until 2012.
Danny salutes Admiral Baldwin, which the admiral returns, when they are both in civilian clothes. Both a Naval Academy cadet, and certainly an admiral, would know that saluting out-of-uniform is disallowed.