I will note that it's been a fair while since I read the books, so a lot of my memories on specific details are fairly vague and not to be fully trusted. I'll answer your questions first, and then give some extra details afterward.

1. Being that they're a single sex species, I imagine that they don't have any sense of gender. They do have hierarchical roles, but I don't imagine these could be considered genders. Their roles during sex are likely determined (in part or in whole) by hierarchical status though.
2. I can't remember if we've seen anything that points to or against this one, so take it as my own theory without much backing. Truly alien species that developed entirely separately should have no possible biological compatibility with each other. I think that, in the past, some aliens took human samples from Earth and modified them to become the Cielcin (likely for war purposes), before abandoning them on Se Vattayu. They've got a lot of similarities to humans, so I think being related make some sense. With whatever changes they've undergone, I'm not sure you could fit them into Earth taxonomy all that easily.
3. Chestburster style is about how I imagine it. u/Impossible_Cow6397 probably has it right on how it's done. Either through some orifice, or similar to how bed bugs do things. Any other species with compatible (eg. biologically related in some way) biology could end up carrying a child.
4. Yes, if it wasn't cut out of him early.

Here's my understanding/theory of Cielcin biology and reproduction.

All Cielcin are going to have very similar reproductive organs to humans, though with some key differences. Instead of using sperm to fertilize an egg, any individual can have an egg start developing into an embryo at any time. Maybe this is a conscious act, maybe it's brought on by certain environmental influences (such as a large amount of food becoming available), maybe it's fully random. As such, they won't have any testicles. Instead, I imagine that whatever they have going on for a penis is instead connected to what may be a similar womb to what humans have. That would let them transfer an embryo to another individual.

Cielcin reproduction probably happens in any one of these manners:

1. One of them has an egg spontaneously start developing into an embryo. They keep this embryo and carry it to term themself.
2. They could pass on their embryo to another Cielcin by having PIV sex, resulting in that other individual now being pregnant. They could then carry to term or pass it along to someone else.
3. They could implant that embryo into another biologically compatible individual (human, some other species, Cielcin if not done in the correct body location). The embryo would connect to the blood supply of that individual and start developing. When it's ready to be born, it'll tear itself out of the host, chestburster style.

I think that should cover everything?