The Cast Showoff: Numerous times the entire main cast gets to show off musical talents. The best is in "A Nightmare on Dick Street," wherein Harry's dream is an elaborate Broadway-style song and dance number that was shot almost entirely in one take.
August sings a song in the episode with Tommy forming a band. Shay Astar, who played August, is now a professional singer-songwriter (though she sang only as a hobby at the time). Astar became a star!
Mary's lackluster but enthusiastic vocal talents are shown off at least as much as the Solomons', despite being poor enough both in-universe and on-screen for a season 3 episode to revolve around her self-recognized lack of singing talent.
Dick: "Nina please, we're talking about prejudice here, something you know nothing about." Nina: "I'm black." Dick: "Oh, right." *grabs a Post-It note and begins writing* "Nina...is...black." *slaps the note on her chest and turns back around*
In the same vein as the keep away, the event Don describes as "Juggling newborns." It's not as bad as it sounds, but it was severe enough for Don to give the Solomons a pretty severe What the Hell, Hero?
The "Mission Song" from the last moments of the final episode is all of these, made all the better by the entire quartet actually knowing how to sing and harmonize reasonably well. Also counts as a Crowning Moment of Funny, Heartwarming, and Awesome.
During one episode, Dick comments that Easter Island was a practical joke that got out of hand. Many listeners will simply associate this joke with the massive stone heads and laugh, but a person who has read about the ecological and societal collapse resulting from overuse of natural resources due to moai construction will understand the "got out of hand" differently.
Trying to blend in as a human physics professor means Dick does equal parts Genius Bonus and alien Techno Babble, like "grading on a transient loop".
When Sally gets a tomato plant and becomes obsessed with him, she names him Jeremy, claiming that she originally wanted to call it "Lycopersicon Esculentum," but that sounded too snotty. That sounds like alien Techno Babble, but is actually the scientific name for tomatoes.
Harry ran for the Rutherford City Council district 9, in "Dick The Vote".
Idiot Plot: There are numerous plots where the problems/misunderstandings would be solved quickly if only some more specific language was used. Of course, if that was the case, things wouldn't be nearly as funny. It also should be mentioned that the idiot plot is justified for this show, since the Solomons are learning Earth culture and of course wouldn't know about many facets of humanity.
Strawman Has a Point: When Dick is called for jury duty, he's entirely gung-ho about the process until he learns the defendant is someone he's met. The case is iron clad and he's completely guilty and Dick knows it, but Dick can't bear to send someone he knows to jail, and have him know Dick in turn. Dick even uses the words "now he has a face" to explain why it bothers him. He's still an idiot, but he sees the defendant as a person and not just a criminal to be convicted.
Legally, he should have been excused from the case anyway, since no one in the jury is supposed to know anyone involved in the case.
Take That, Scrappy!: August gets her comeuppance. After they break up Tommy convinces everyone that he dumped her which constantly gets thrown in her face.
Tear Jerker: 3rd Rock is a relentlessly goofy show, which sometimes makes the rare moments of drama that much more striking.
Season 4's "Sally Forth" ends with Sally and Don's breakup in Don's car, after his marriage proposal goes south.
The final episode, "The Thing That Wouldn't Die, Pt. 2," where the Solomons reminisce over everything they'll miss about Earth, leading to Mary realizing at the last second that she just can't leave her life behind for Dick.