(at about 13 minutes) When Beethoven steals Billy's cookies during their first encounter, Billy is wearing black jeans. A few moments later when Beethoven knocks Billy down and is licking him, Billy is wearing beige jeans. After Beethoven gets off Billy, Billy is wearing black jeans.
In the scene when the boy names the dog Beethoven, the orchestra on the TV is indeed playing Beethoven's Fifth, as the announcer says. However, when the boy's father enters the living room and hears a classic melody of Beethoven, what you hear comes from Beethoven's Symphony No. 9, not Fifth.
(at about 25 minutes) There are 5 animal carriers on the trucks tailgate. 3 on the right and 2 on the left. The next shots show only 1 carrier on the left.
The pizza that Beethoven discovers on the ground before he gets captured is obviously fake.