Should we gather the "just to clear it up" statements all in one place?

"Xenomorph" is not any sort of official in-universe term for these things, though I believe Alien: Romulus at least does have scientists calling it "xenomorph <some number>" indicating it has been catalogued. Xenomorph is just a generic term for a non-humanoid alien, though, what PFC Hudson called a "bug" when the term was introduced as Lt Gorman was briefing the other Marines about what they might expect. This wasn't their first time encountering hostile life and they called a lot of it by this name.

The xenomorphs (as fans have taking to calling it) are not silicon-based, but rather some blend of organic and inorganic. They're bioengineered superweapons, not naturally occurring. As far as we can tell, they don't eat at all, at least in their adult form, as they grow to full size in a matter of hours without us ever seeing them consume anything. They only seem to need hosts in their larval form. This is also why they mostly don't kill the humans they attack unless they have to because they're under threat. They instead attach them to a resin they secrete, where they are used as incubators for more larval forms. Their entire purpose seems to boil down to multiplying as rapidly as possible. I believe Ridley Scott has suggested they consume electricity or radiation in general directly, rather than extracting energy from food.

They were not actually created by the "engineers," as Shaw from Prometheus called the space jockey guys. They were created by David, the first synthetic sentient android created by Weyland, when he and Shaw hijacked an engineer ship. He returned to their home world, released the black goo and killed everything, and experiments with its ability to create multistage lifecycle semi-organic lifeforms, which he eventually managed to turn into the xenomorph as it was seen in the original Alien.

The black goo itself seems to have been some sort of quasi-religious experiment thing the engineers developed billions of years ago that has the ability to dissolve organic matter, attach itself to DNA, and rewrite it to become some kind of more "advanced" lifeform. The beginning of Prometheus shows them doing this on earth, with a sacrificial engineer dissolving himself into primordial water to seed what would eventually become all earth life. This doesn't imply earth-life is genetically similar to the xenomorph, but rather to the engineers, who were humanoid. So Ridley Scott (or Damon Lindelof or whoever) is implying earth life was meant to eventually become humanoid all along, like we're the ultimate end of an experiment in engineered evolution.

However, back on their home planet, they didn't just stop doing new things for four billion years. The black goo of the present in Prometheus was similar to the original used to seed earth, but seemingly had changed at that point and specifically created more xenomorph-like things, now by infecting other lifeforms and incubating in them, rather than dissolving them into free-floating DNA.

All of that addresses whether xenomorphs are "related" in any biological sense to life on earth, but either way, I'm not convinced arthropods are the right analogy. The reality of their size and movement is obviously confounded by the fact they're played by humans wearing costumes, but I'd be surprised if they didn't have endoskeletons. Ridley Scott has stated the original intention of the creature design was to invoke armor. They're insect-like in appearance, but arthropod exoskeletons are not really armor. In fact, arthropods very rarely grow as large as humans, with the only examples all being extinct, and they had paper-thin exoskeletons, in part because molting something that large is difficult and expensive. To me, I think they're more similar to certain types of dinosaurs with hardened exterior armor. That said, they are clearly shown molting, but the fact they have claws and teeth seems to suggest they probably also have bones. I suspect this is mostly just confusion on the part of the designers or the simple fact they aren't really trying to make something that is biologically coherent or plausible so much as a nightmare made flesh.

I don't think it's ever made clear what the engineers were trying to do. The timelines are a little jacked up, but the derelict the Prometheus crew discovers can't be the same one discovered later by the Nostromo, because the one of the Nostromo has actual fully-realized xenomorphs, which the narrative eventually makes clear were created by David after the events of Prometheus. The derelict in that case was simply carrying black goo, but a weaponized version, and carrying it to earth, apparently intending to destroy all life there. Why is never explained, nor does it seem to be the case that this was some civilization-wide decision, since the mission failed and no follow-on was ever launched. This could have been some rogue faction that weaponized black goo and attempted to destroy life that had earlier been created by the non-weaponized older version out of disagreement with the larger historical goals of their own people.

Weyland-Yutani corps by the time of the mainline films seems to not understand what they are dealing with. They understand they have found some substance that can meld with organic life and create a stronger version of it that doesn't need food and can survive in a vacuum. Romulus seems to suggest that the goal at that point was to create a race of hybrid mine workers that would survive the harsh conditions better than humans. They don't seem to have ever realized that David had created the xenomorphs as they existed at that point and his goal was not to create better humans but to destroy all other life completely and replace it with xenomorphs.

Considering all I've written to this point, I'm amazed at how silly the mythology of this series eventually got after starting as a simple great horror flick that needed no explanation or backstory.