When talking about orbiting bodies it is important to keep the timescales in mind. At these timescales everything is practically a liquid. Astronomers can generally do a pretty good first-order approximation of the shape of large objects by just pretending it is made of pure water. The reason is that celestial mechanics exhibits such strong forces over such long timescales that even the strongest materials tend to flow.
So, if you want to know what shapes you can make in space, just imagine what a large drop of water will do.
A post above asked why skyscrapers can resist gravity when the Earth can’t. The answer is that skyscrapers DON’T resist gravity on celestial time scales. From an astronomical point of view, a skyscraper is like a spout of water that is falling and will very quickly enter equilibrium.
But to be less flippant - there’s a reason concrete buildings can only be so high - because there’s a limit to how much weight a lump of concrete can support before crumbling, even when not affected by weathering, freeze-thaw cycle, rusting rebar or salt water. The same limit applies to steel pillars - just taller. In geological scales, tens to hundreds of miles high and millions of years long, yes you are right, everything flows under pressure. Glaciers are just the more blatant example of the principle.
This is generally the case. A rapidly spinning body tends toward a lenticular shape (oblate rather than prolate). However, it appears that Haumea has satellites, which affect the shape of the body. If a rapidly spinning body has a significant satellite, that satellite will participate in tidal interactions that will, over time, pull the satellite into a synchronous orbit, pulling the primary into prolation. If Haumea is in fact a rugby ball (I suspect our imaging equipment is not adequate to resolve its actual shape at this distance), this is almost certainly what is going on there. In several million years, the tidal forces may be enough to spiral the satellite into the primary, resulting in a merging body that initially looks kind of similar to Arrokoth but, after a few thousand years, will settle into a lenticular shape due to its increased spin.
Mention should be made of Hal Clement’s Mesklin, the setting for his novel Mission of Gravity. There have been illustrations of it depicting it as a severely flattened spheroid, although I understand that some have taken issue with the depiction (and I’m not sure why)
The Ring would never transit its sun. From any given vantage point, it’ll block some percentage of its sun’s light (0% from the vast majority of vantage points), but that percentage will stay constant from any given vantage. You might still notice it if you’re doing deep all-sky surveys from multiple worlds, and at least one of those worlds happens to be lined up with the Ring: Then you’d have one star out of umpteen-many that appeared dimmer in one star catalog out of many. It could easily take a very long time for anyone to even notice or care, and longer to rule out instrumental error or any other mundane explanation.
The Ringworld would be a pretty bright object, a million miles across and more than a hundred million miles along the long axis, with an albedo similar to Earth. It would far brighter than a typical gas giant because of its size, and if the angle was favourable this object should have been detectable at interstellar distances. Assuming reasonably large telescopes in Known Space, someone should have seen it hundreds of years before Wu got there.
As I noted, you’d see the dimming only if looking at the system edge on (although the 1.6 million km width of the ring makes that a not inconsequential likelihood; assuming it is the same diameter as Earth’s orbit around the Sun, the possibility of the ring possibly occluding the star is about 0.7%) but you will otherwise see light reflected off of the surface in an odd, crescent-like distribution. About the only way you wouldn’t see some evidence of the ring is if you were actually looking fairly close to axis on, where the incidence of reflected light is obtuse.
And although it is widely noted that the Ringworld is statically unstable, but it will also be prone to both torque-free precession and almost certainly both free and forced nutation. In fact, there is a whole range of dynamics that a ring of that size would undergo even assuming it is perfectly rigid that the circumferentially-mounted reaction ramjets would have great difficulty correcting so we can expect a certain amount of lateral wobble. The fusion-powered ramjets themselves would have very characteristic spectral lines quite distinct from any natural stellar processes, so even with all of the corrective actions in The Ringworld Engineers, the structure is really not feasible.
We can assume that the collective humanity of Known Space, with its familiarity with multiple aggressive alien species, would be actively surveilling star systems for hundreds if not thousands of stars beyond the bounds of explored systems for signs of any potentially threatening new species. We are not told exactly how far the Ringworld is from human space but it is near enough to be reached with a conventional hyperdrive capable of flying only a couple of lightyears per ship’s day so it must be within a few hundred parsecs and of course would have to be close enough for Pak Protectors to have sampled Kzin and other worlds and returned the species to the Ringworld for purposes unknown.
This is the problem in general with big dumb objects (BDO) in ‘hard’ science fiction; the more you explain them, the less plausible they become, because authors tend to think of them in human purposes, e.g. creating a giant ring for living space and agricultural use even though it is essentially indefensible from attack or natural hazard, and an advanced species would probably move beyond natural agriculture to feed their interstellar civilization anyway. Yes, for Paks, protecting and expanding their breeder population is the evolutionarily hardwired into their instincts superior to all other purposes, which again reflects a superficial understanding of evolution of behavioral characteristics.