Revealing mistake: When the master's gold falls into the water off his boat, several of the gold items can be seen floating, something no gold item should. (00:40:00)
Revealing mistake: When Gandalf shakes Bilbo's hand it should be a lot bigger than his but it is the same. (02:25:26)
Revealing mistake: During the battle of Dale, Bard confronts Alfrid, who is dressed as a woman and is trying to escape with looted gold. During the confrontation a troll climbs over the parapet and is killed by Bard. As the troll falls, Alfrid also falls. When he lands, the ground near his hand ripples like fabric covered in sand.
Answer: Aragon is 87 in LOTR: Two Towers (as mentioned in the extended edition). The Hobbit takes place about 60 years before that so Aragon is 27 at the time. Sauron is banished to Mordor in Battle of the 5 Armies. You see his "spirit" flee and the White Council, mention he went back to the east. Between Sauron and Sarumon, it's mentioned more in other LOTR books, but Sarumon (while still technically good) has been looking for the One Ring. He does not join Sauron as much as uses the alliance for his benefit hoping to find the ring first.