As for the backstory: Originally, all three siblings were destined to "die" from malnourishment, not necessarily of food or water, but love, as per Yuri and Tabuki's comment about unloved children, and the entire idea of the Child Broiler. Notably, "Kiga, " which appears on the apples, means "hunger/starvation." For Shouma, there is an aspect of parental neglect and abandonment on the part of their actions and the consequences (this situation would then apply to #2 always consuming food).
But then Kanba got an apple, "The Fruit of Fate." He shares it with Shoma, saving them both. Shoma shares his half with Himari, saving them both, but the 'curse' or 'punishment' for sharing is that Himari must die eventually. In the end, Shoma returns his half to Kanba, who gives it to Himari who already has his half, saving her (a full apple) but he dies since he has no fruit anymore, becoming “invisible” and breaking apart into glass like in the Child Broiler. In the meanwhile, Ringo uses a spell to transfer fates. However, the cost of the spell is being burned to death, which Shoma steps in and takes instead, since he was about to die anyways from the lack of a fruit. So Shoma and Kanba die as a result, and Ringo and Himari live.
So basically, the Penguindrum is life/love, and throughout the whole show the brothers have been trying to give those things to Himari, and in the end do so and come full circle. Thus the title comes into play, as the Penguindrum "rotates/spins" through all three.
A purposeless Shouma saved Himari and gave her a family, thus shared his apple with her; Himari was henceforth given a purpose, but keep in mind that Shouma received nothing in turn. Then Kanba gets accepted into the family and Himari gives him the bandaid to keep the pain away, thus Kanba decides he'll do anything for her; here Himari was unknowingly sharing her apple with Kanba, because she gave him a purpose (and the bandaid appears on #1). Kanba was able to actually receive the apple because he wasn't completely empty and devoid of purpose like Shouma himself was, as Kanba already had precious people he wanted to protect (Masako and Mario, his words in the cage are a reference to them) as such his "box" was able to connect to the outside world, if only for a moment and the apple reaches him. Now, this changes with Himari, "going genes/instinct" not in incest, but by choosing her over his biological family, and Masako wants to reverse that (thus Esmeralda always taking Threetie's wigs). In the end, Kanba gives back his whole apple/purpose, and so succumbs to his fate.
When Kanba shares the apple with Shouma, Shouma's purpose becomes precisely that: Himari no Tame Ni + Kanba = Takakura Family's wellbeing. That's the reason why Shouma is so single-minded about keeping the family united; because thats' what gave meaning to his existence, as such was what he inherited that day Kanba shared his apple with him. He, however, still remains kind of empty and from that comes his apparent uncapability to love, but Ringo comes around to change all that. When Ringo (whose name speaks for itself) comes around, she effectively becomes an apple for Shouma, a "whole" one all to himself, and she fulfills the part of Shouma that had remained closed-off to the world. He isn't able to accept it however until he lets his barriers down. The key difference here is that Ringo's metaphorical apple didn't give him a purpose, as he already had one, but gave him the will to break the chains of fate and be assertive, which is what he lacked.
Seito Sakakibara is a pretty blatant influence on Sanetoshi as a character too. He uses a lot of the same terms and phrasing as Sakakibara did in his creepy letter to the TV station. Sakakibara thought of himself as an "invisible entity" created by society (he calls out public education specifically) and that he had conversed with another such entity, the only other one like him. He claimed that he took credit for the murders he committed, that he otherwise could have gotten away with, to make the world recognize his existence.
The Sanetoshi - Aum Shinrikyo/Shoko Asahara connection seems pretty clear. The parallel to the actual subway attacks done by a fanatical group bent on "saving" society/others aside, there are other things. Seizon Senryakyu/Survival Strategy could be a play on Aum Shinrikyo's own "Perfect Salvation Initiation" (PSI) method, or their PR slogan which was "Aum Salvation Plan, " which promised to "help people with no direction or sense of future to redeem themselves" (aka, "those who will never amount to anything.")
Also- Asahara, the cult leader, was described by his followers as perfect, spotless. Because the souls of ordinary people are "covered with dirt. they are polluted, but Sonshi's (Nakahara) soul is different from ours. It is open, uncovered, without any dirt." A key point of Sanetoshi's character design is his immaculate white. He also presents himself as a doctor, same as Asahara who came from a medical background.
Sanetoshi is an embodiment of spite and hate, while Momoka is an embodiment of forgiveness and love. At the very core the opposition between Sanetoshi and Momoka is, simplistically, selfishness vs selflessness. Sanetoshi doesn't give a shit about harming others to satisfy his desires and uses the same ideology to win over followers (i.e. Kanba) to the same methods. He doesn't kill to end others' suffering, since that would require giving a shit about other people––Sanetoshi wants to kill and end the world because he doesn't like it, because he's invisible, and so on.
Obviously, combined with his imagery and behavior, he is effectively Satan, even moreso than Akio from Utena, and compounded by the role of the Black Bunnies as a Serpent role in Shouma's Mary's Little Lambs story due to its Garden of Eden parallels. Furthermore, the Apple/Penguindrum is shared love and purpose in life, but also shared punishment, and in the same way Adam and Eve shared the punishment of mortality (living is punishment according to Kanba), but was also given the solution out of punishment, by love
The box metaphor is all about selfishness and self-preservation (survival strategies). Ultimately, the fate Sanetoshi is talking about––the "self-serving rules" that govern the world––is the cynical idea that people are selfish and––not even will not, CANNOT help each other because it's contrary to their best interests as dictated by "fate" (genes/instinct/whatever). Kanba sharing the fruit with Shoma when they were trapped in the cages effectively broke fate because he violated that principal. Thus, the boxes are fate, or what could be seen as the human condition itself.
Fate is not supposed to be taken as a fact or a concrete force at work determined by divine will, but something else. Fate isn't used in the classical sense here, necessarily— fate (especially in relation to unwanted children) is meant to be the future that people resign themselves to as a result of the selfishness of others (meaning those stuck in their boxes). The future that dictates they will never amount to anything. Altruism is what changes 'fate', or the bleak and awful future so many children face as a result of the lack of love. That is, genuinely reaching out for an unwanted person and making them a part of your life, make them feel needed, and giving them the power to do the same for others. To step outside your box and truly reach out to another person who needs love will change their life completely.
TL;DREpisode 1: "This is all about love."
