Arranged Marriage: Rhea tries to legitimize her invasion by having Mon-El marry Lena and produce an heir, whether they want to or not (using a Uterine Replicator, since their answer is "not").
Artistic License – Law: The President admits that she was born on another planet, thus her presidency is in violation of the Natural-born-citizen clause. Of course, since nobody knows this, she may just have (very, very good) fake documents (which is still illegal).
Winn "accidentally" bumps into Hank, who predictably responds in a very violent manner. While this initially appears to just be a sight gag, it later turns out Winn placed a device on him that comes in handy later.
Superman is initially contacted, but he's absent. He reappears in the end, under Rhea's control.
Eventually, Kara realizes that the only way she can get on to the ship to save Mon-El and Lena before Alex blows it up is to join forces with the genocidal terrorists she has been at war with for months now, and who have her father captive.
Also, CADMUS believes that all aliens should be booted off Earth or killed, and that superheroes are not needed—Earth can take care of itself, thank you very much. As such, working with the alien superheroes in the employ of an organization that fights evil aliens but protects the friendly ones as well as humans is very much the same kind of last resort for them.
Hostage Situation: To get Mon-El and Lena to go along with her plans, Rhea threatens to blow up hospitals in National City, starting with a children's hospital Lena helps fund.
Huge Holographic Head: Rhea's mothership projects her head above National City to broadcast her intent. Later, Cat and Winn hijack the device, displaying Cat's own head to deliver her resistance message.
Idiot Ball: President Marsdin flies Air Force One right at a hostile alien ship that is far more powerful than a few fighter jets and a passenger plane, and the only reason they seem to do this is to have an excuse to drop Cat back inside the city and reveal Marsdin is actually an alien.
President Marsdin is revealed to be an alien to the main cast.
Cat learns that James is Guardian. On the spot.
Kara finds out that Rhea killed her husband.
Lena finds out Mon-El is Rhea's son, though it's unclear if she remembers him from their brief meeting.
Jerk with a Heart of Jerk: Lillian teams up with Supergirl to save Lena out of genuine concern, then throws her and Mon-El under the bus at the last minute and treats Lena as ungrateful for protesting.
Militaries Are Useless: The world (well, National City for now, but surely the rest of the world is next) is threatened by an Alien Invasion, and the only military response we see are two fighter jets that are blown out of the sky in a matter of seconds, and they were only there to escort Air Force One. What, no BVR missile strikes, no drone attacks? It seems the entire US military is banking on a single Wave Motion Gun to save the day.
Modest Royalty: Discussed. Cat Grant says that a real queen doesn't need tiaras to show they are.
Never My Fault: At the start of the episode Rhea claims she's been taking care of Lena since Supergirl knocked her out; thankfully Lena remembers everything else that happened.
Not So Different: Supergirl teams up with a delusional, xenophobic, psychopathic Evil Matriarch who refuses to accept that her child doesn't share her viewpoint to fight a delusional, xenophobic, psychopathic Evil Matriarch who refuses to accept that her child doesn't share her viewpoint.
Paper-Thin Disguise: Cat sees right through James's Guardian disguise, since she can still see his eyes.
Pragmatic Villainy: Lillian explains to Supergirl that she has yet to reveal her secret identity to her daughter because she fully believes Lena will figure it out on her own and will hate Kara all the more for lying to her.
President Superhero: The president is an alien who can shapeshift and survive an exploding plane.
Properly Paranoid: Rightly suspecting that CADMUS will betray them at some point, Winn slips an override device on Henshaw that hijacks control of him when Lillian tries to leave Supergirl and Mon-El to die.
Reality Ensues: Giving a supervillain bent on world domination a "Reason You Suck" Speech twice can end badly, as Cat finds out. The first time results in Rhea blowing up Air Force One, and the second time leads to Rhea sending a hit squad on her.
Shout-Out: Supergirl calls Henshaw R2-D2 for his ability to interface with the computer on the mothership. The port he interfaces with even rotates back and forth like those in Star Wars.
Skewed Priorities: Cat doesn't mind the president being an alien or that the president endangered her life, but she better still be a Democrat. Also, she's more annoyed that her old office now has a gym and sports paraphernalia than that it's been trashed by an alien kid with Psychic Powers. Also doubles as Comically Missing the Point.
Take That!: During her Rousing Speech to the people of National City, Cat mocks Rhea's promise to "make our world great again," a riff on Donald Trump's "Make America Great Again" campaign slogan.
Title Drop: Done courtesy of Cat Grant when encouraging everyone to fight the occupation.
Unexpected Character: It was announced that Superman will only show up in the Season Finale. What wasn't announced is that the final minutes of the episode will show exactly why he's involved.
Uterine Replicator: The Daxamites have the technology to grow children using DNA samples from the two prospective parents, skipping the middleman. Rhea plans to use this to make an heir from Mon-El and Lena since they aren't going along with it willingly.
Wave Motion Gun: The positron cannon, Earth's final defense against the Alien Invasion, is housed on the DEO rooftop. It never gets a chance to fire.
We Come in Peace — Shoot to Kill: Rhea tells the people of Earth how she's come to improve their lives as her ships lay waste to the city and her soldiers beam down and attack everyone in sight.
Wham Shot: Just as the positron cannon is about to fire, Eye Beams from offscreen trash it. Then Rhea boasts that she isn't beaten yet and Supergirl is attacked by Superman.
Women Are Wiser: Invoked and satirized by Cat Grant when she's mediating between Rhea and Marsdin, saying she's seen enough petty squabbling and power posturing from the politicians in DC and she thought she'd get something slightly more impressive from two women. It works somewhat to soothe their egos, but not enough.
Would Hurt a Child: Rhea threatens to blow up the Luthor Children's Hospital to force Mon-El and Lena's cooperation.
Your Princess Is in Another Castle: Alex is ready to blow up Rhea's mothership with a positron cannon, Mon-El and Lena are safe, Henshaw and Lillian are contained, and all that's left is for Supergirl to try one last time to talk some sense into Rhea. Unfortunately, we still have an episode to go, so she pulls out a trump card in the form of Superman, who's on her side now.