1d4chan is not any kind of primary source; Hashkar is a petitioner, yes, but the notion that he is a petitioner of the Lady is a rumor, as indicated in Faction War. Furthermore, it is entirely possible to be a petitioner without the anchor point being a deity. The Lady of Pain is not a god of any kind.
Unless there's a divine realm they belong in, they generally don't.
Long road. Through the Ethereal to its nearest Astral contact point, then into the Astral to get where they belong. If there is no valid contact somehow, they get stuck (this is what happens around Athas). It should be noted that the elemental nodes of Athas do have conduits to the broader Elemental Planes, so if you die while in one of them, you should get out just fine.Let's say that an adventurer is killed on a demiplane or elemental plane, connected to the Ethereal but with no Astral conduit. What happens when they die?
Definitely not.Do they get (hypothetically) get sucked up through the Ordial Plane?
We've done that one before; short answer is that ghosts resist the pull of the Astral by anchoring themselves to the Ethereal, and until they're willing to release that anchor, they can't pass on.Likewise, how do Ghosts come to manifest on the Ethereal, rather than the Astral? They are a being of pure soul; how do they manage to anchor themselves in the the plane of substance rather than belief?
Not tremendously common, and as noted they would be involved in genealogy and history, recordkeeping and scholarship. Less community-oriented than dwarven clerics, more behind-the-scenes. Dwarves have long memories and would archive the truenames of enemies as secret weapons that can be wielded when absolutely necessary, since truenaming is, you know... mechanically abysmal.
In general, gods who share a portfolio cannot share territory; syncretism or redefinition can result, both unfortunate outcomes for a deity. Nobody wants to be the sun god who represents just the dusk and nothing else, but that tends to be how such things end up going - one god gets treated as a more specific, specialized, and lower-scope power than the other, with commensurate impact on worshiper base and, given a large enough timescale, personal nature and abilities.
No.
Functionally true neutral. Neither of them has a use for being measured in such terms.2. What's Ao and Lady Of Pain alignment?
Oh each plane has lots of unique local wildlife, but not entire exemplar races in the fashion of the main nine. I considered the idea, as Tzardok pointed out, but ultimately didn't care for it.
It can be difficult at times, but there are a few things that would set them apart - petitioners (with few select and specific exceptions) cannot learn, change, or grow, and often display a singlemindedness to their existence. They also have outsider traits, do not require sustenance or sleep, are planar committed, and possess very little ability compared to the living (being reset to functionally 1st level permanently). One of the most evident traits, however, is that petitioners, while generally aware of their identities, are a bit mentally checked out from them - you'd need to remind one to think about having had a name or a life.2) For some planes, I can't really understand the difference between petitioners and mortals residing there. Specifically Bytopia and Acheron. Most of Bytopia residents are gnomes and dwarves, mortal races that love this place and also petitioners have similar forms and even similar life style. Pretty much the same is with Acheron, but with goblinoids instead of gnomes. So, if someone visits one of those planes, how he would distinguish between a mortal resident and a petitioner?
Very very limited, there... I suppose the Planescape Monstrous Compendium II would be your best source.3) Rilmani have been mentioned quite a lot here. But if finding about other outsiders is relatively easy, I don't have any info on those. Where I can read about them? I don't mean stats, I am much more interested in lore, like how they usually behave, how they are organized, how their government looks like, how they treat outsiders, what relation they have to Blood War etc.