That orb/halo forming above Chamber's head when it levitates? It's a gravitational singularity that pulls the mecha (and other nearby objects as shown on a few occasions) towards it; if the Galactic Alliance can create wormholes, presumably with Artificial Gravity (The only force theorized to be able to create rifts in the fabric of time-space), it certainly wouldn't be beyond them to use it for propulsion. Similar drives are described in Star Carrier, for example.
Whalesquid, a.k.a. Hideauze, are considered holy by Earthlings. Why? Because they were originally created by humans to help humans adapt to space by becoming bodies for them via genetic engineering. How the Hideauze are treated makes perfect sense if you consider them people that can survive underwater.
The archive footage shows that the evolved humans were highly respected and revered for being pioneers in an endeavor that might have saved the human race. This respect must have been retained over time and evolved into worship.
Or it could just be simple pragmatism, given that whalesquid are apparently horrendously dangerous to fight.
The reverence is clearly cultural; the Gargantians aren't actually very afraid of them, so their borderline worship of whalesquid has to do with more than pragmatism.
The naming of the Hideauze instead of "Evolvers", as they were known on Earth, is actually quite telling. They're hideous in the eyes of the Galactic Alliance. A few generations of propaganda and linguistic shifts....
Chamber claims that a random glitch in the wormhole sent them to a completely random location, which happened to be Earth. One troper estimated this to be a 1 in 10^39, i.e. one in a trillion billion billion billion chance for that to happen. However, not only did Chamber end up on Earth, but Striker did as well, which implies that it wasn't a random chance. The Continental Union built a wormhole gate near Earth similar to the one employed by the Galactic Alliance. When the Galactic Alliance's wormhole collapsed, it's likely that both machine calibers fell through space time toward Earth because there was a preexisting "fissure" located there where the old wormhole at Earth used to be.
The Wham Episode reveals to us that the Hideauze are actually genetically engineered Humans. This is actually lightly foreshadowed in episode 1. If you pay careful attention during the initial battle you can catch blink-and-you'll-miss-it shots of the Hideauze with disturbingly human(or at least simian)-like faces behind the mouths...◊
In the final episode, why on Earth would Lukkage develop feelings for Pinion? Well, they both want power. They're both better people than they appear. They both have big egos. And they both have great hair.
You'd think that displaying Hideauze/whalesquids with humanlike teeth is solely a way to foreshadow their status as Transhuman Aliens. You're wrong. Their appearance, and perhaps the very concept, is inspired by this very much real cephalopod.
The series might end well for Ledo and Earth Humans - for the time being, but the Galactic Alliance of Humankind and the ruthless space Hideauze are still out there somewhere. Not only is the Alliance said to be always on the lookout for habitable planets, there's also Ledo's distress signal which Chamber claimed would reach them in a few thousand years. It may take a really long time, but eventually they'll find Earth, and it's hard to imagine a happy outcome when that happens.
Then again, a few thousand years ago both the Galactic Alliance of Humankind and the Hideauze were Earth Humans. But all they've done with their time is refine their methods of killing each other at the expense of everything else. Now that Earth humans and whalesquid have started rebuilding Earth in earnest cooperation, they should be capable of handling them within a decade or so. Though the sequel OAVs imply they may not have that much time...