Not an eps, but TSFS can cheer me up.
At first, it seems unlikely. The movie is quite a downer.
It starts with mourning for Spock, then medical problems for McCoy, then forced retirement for the Enterprise, then the Genesis Planet becomes a disaster, David Marcus is killed, the Grissom is destroyed with its crew, the Enterprise crew sacrifices their careers and possibly their lives, then the Enterprise is destroyed... it's bad news on top of more bad news all thru the movie.
And yet the crew never gives up despite any obstacles, all for their friendship and for Spock. That's tight. Against all adversity, to see love and friendship like that cheers me up.
ST III is a comfort for me too. In addition to the elements you mentioned, it also reminds me of the era it was made and released, and I have a fondness for that time of my life.
I would also say the teaser trailer for ST VI is a good little bit of Trek-ness. The only time the films ever specifically referenced the classic series on screen. That trailer is just great, and the sign-off at the end of VI is heartwarming to me as well.