It takes many listens for it to fully settle in that Andy, Akiva, and Jorma, when 13 years old, got drunk off Martian wine and then proceeded to have sex with a male alien at the same time so that they could repopulate a dying alien race with their sperm.
And while the alien did initially leave the trio with much MC swag ("Jack Sparrow", and "Turtleneck & Chain"), the whole event apparently left the boys with varying forms of penile dysfunction ("Jizz in My Pants", "We're Back", and "I Just Had Sex"). This results in their failure to find success with beautiful women no matter what, even when the ladies themselves try to take the initiative ("Attracted to Us"). Out of desperation, the guys aim for something much lower on the scale ("Dreamgirl"), but that also fails. The lack of any decently meaningful relationships in the boys' lives turns them certifiably insane at one point, and they proceed to lose whatever street cred they had ("Who Says We're Wack?") by doing generally insane things ("Trouble on Dookie Island", "Punch You in the Jeans", "The Creep"). Samberg soon develops extreme bi-polarity that leads to a lifestyle of constant self-mutilation ("Like a Boss") and violently rejecting every attempt at kindness around him ("Threw It On The Ground"), while Akiva and Jorma kinda settle on being dorks with no real social skills ("We Like Sportz"). Eventually, the trio realize that they need to deal with the issue once and for all, but that also ends up failing to produce any real favorable results ("After Party").
The Boss's crazier actions (like chopping his balls off and bombing Russians) can be traced all the way back to getting rejected by his co-worker.
Everything from "crash my car" onwards can be explained as hallicunations from a combination of drugs & alcohol, emotional stress and injuries from falling out of a window and... well, crashing his car.
The first is the "straight interpretation" (pun intended). Taking everything at face value, the two men want to have sex with the same woman, who requests that they participate in a three-way. They then get carried away during the act, but justify it by invoking an unspoken rule about sex, that states that anything done in a threesome doesn't count as gay.
The second builds off of the setup for the first line. Despite just leaving each others' mothers' houses after having sex, the two guys feel unfulfilled and jealous that everyone else in the city is having sex but them. They decide to go after a girl closer to their age, but when they find that they're pursuing the same girl, they're too desperate to turn down the threesome, and try to come to terms with what happened while they were in bed together by inventing "The Golden Rule".
The third, implicit in this video as well as "Motherlover", is that the two guys are secretly attracted to each other. When the woman asks for a threesome, they agree, and insist that it's not gay because of her presence, even going so far as to convince themselves, but when the lights are out, they legitimately enjoy the experience of being in bed with one another. As soon as the lights come on, however, they revert to claiming that they're not gay because Lady Gaga was there for part of it.