Mandy Patinkin was dressed in warm clothes during the final scenes because he was going to perform a concert in New York soon after and was afraid of getting sick. The temperature was 22 degrees Fahrenheit (-5 degrees Celsius).
Tobias Hankel's former drug program sponsor tells Prentiss and JJ that after his wife left him, Hankel's father "went Section 8." Section 8 is a category of discharge from the United States military used for a service member judged mentally unfit for service and spoken in slang as a way to indicate that a person is crazy. The term was popularized by its use in M*A*S*H (1972), in which Cpl. Maxwell Q. Klinger regularly wears dresses and other outlandish outfits in an attempt to get a Section 8 discharge.
The Bible verse from Ecclesiastes 7:20 quoted by Hotch ("there is not a righteous man on Earth who does what is right and never sins") is also quoted by Reid in Spencer (2017).
When Reid's shoes are taken off, there is a brief glimpse of both of his socks before one sock is taken off. You can see that the socks do not match. This is more than just a character quirk. Matthew Gray Gubler does not like to wear matching socks because bad things have happened to him when he does. So he considers it good luck to wear socks that do not match and bad luck to wear socks that do.