Hunting Xenomorphs is a coming-of-age ritual. And as Lex says, "They're teenagers?" So why is it that the previous Predators weren't always honorable and were sometimes sneaky jerks, their killing apparently pointless and for fun? Because they were teenagers going through puberty, and they'd come to Earth to train for the real thing. They just happen to be pre-pubescents who are seven feet tall.
While the hunt in AvP is explicitly stated to be a rite of passage for new warriors, there's no indication that any of the hunters in the other movies are not adults. The extended materials state that all adult predators go on regular hunts. As for the killings being pointless and the predators not always being honorable, this is a case of Blue-and-Orange Morality — the predators have a clear and rigid honor code that they never waver from, and the hunts are for an individual to acquire prestige and raise their social standing.
There's one reason why the first film ends with many humans dying, save for Lex - it's because it's set centuries before the events of Alien and much of humanity has yet to realize the potential terror of the Xenomorphs long before the events of that film, so it must follow that the deaths of virtually all of Weyland's expedition team should make that very difficult. Adding to that, we can also imagine how difficult it is for Lex, the only survivor of the expedition to Antarctica, to convince other humans about the existence of the Xenomorphs as possibly the very first human to encounter them and live to tell the tale with respect to the larger continuity picture of the Alien, Predator and Alien vs. Predator films released thus far.
They mention early on that the pyramid has markings and structures the same as several civilizations, there is a good chance that this wasn't the only pyramid with an Alien Queen buried due to continental shifts and the like, think about all those civilizations that just up and disappeared one day.
What assures us that the Queen is truly defeated? She was perfectly able to survive thousands of years frozen and the species is shown to be able to withstand the vacuum of space, so she can't drown or freeze to death. Hopefully the chain or the decrepit cistern it's tied to won't break... and that it's a conveniently near oceanic trench with Crush Depth she's being dragged into.
