How do the seeded skaters choose their events?
First, the host countries choose their three athletes/teams. If they have a top-three seeded skater, they can choose that skater. If they have two or three top-three seeded skaters, the host may choose 1 of them. The host country may also set aside spots without naming the athletes who will fill them.
Next, rumor has it, the #1 skater's two requests among the remaining countries are honored. Followed by skater #2's requests. And skater #3 goes to the two open spots. (It could easily be the skater's federation that decides where their athlete will go rather than the skater. Who knows?)
Using men as an example, does Chen choose an event first, then Uno chooses an event, then Kolyada chooses an event, then repeat the process for fourth, fifth and sixth events? Or does Chen choose two, Uno chooses two, and Kolyada gets what is left over?
Chen chooses two, Uno chooses two, Kolyada gets what's left over. (But probably Chen gets picked for SA as one of SA's three host athletes first. And Japan has to decide what to do between Hanyu & Uno for which will go to NHK, provided both skate next season).
Do the skaters seeded fourth, fifth and sixth go through the same process as the skaters seeded first, second and third?
Kinda sorta. Host countries have an order of selection. The first country selects an athlete from 4th/5th/6th, then the next country, then the next country, until the 4-6th place teams have been selected. (Some may have been selected by their own country to skate there prior to this happening). Skaters can let their federation know where they would like to skate. And federations can discuss it or try to make arrangements in advance. But ultimately, you go to whichever host countries select you first during the draw.
Does this happen before any other considerations, i.e. invitations issued to skaters seventh to twelfth (tenth for pairs and dance), host pick assignments?
Just trying to figure out in what order these things happen.
All the 4-6th placed teams must be assigned their events prior to selections for the 7-10th place skaters. Then all of those skaters must be assigned their events prior to any further invitations.
At that point, still going in the order of host-country selection, all non-seeded skaters on the top 24 SB and top 24 World Standing List must be selected for their one guaranteed spot.
After that, host countries may fill any remaining openings by inviting whoever they want. They are supposed to follow some criteria--invite athletes who have the minimum scores and prioritize high ranked athletes on the SB list. But, really, they have a lot of leeway there. They can select athletes who are well known and sell tickets. They can select athletes who train close to home and are cheaper to host. They can invite athletes they think are more likely to lose to their own host athletes. They can make arrangements in advance to trade invitations with other host countries that need an event for one of their own athletes/teams.
Once all the spots are full, all the spots are full.
Last season, at that point, there was a 10 athlete/team replacement list based on the next 10 athletes without 2 spots and who had the highest SB scores from the previous season. If a spot opens up due to an athlete/team withdrawing, the host country is supposed to invite another athlete from the replacement list.
But no one enforces it, and sometimes athletes ranked below the replacement list receive invites. Especially if the opening occurs late and it is easier to invite an athlete who lives close.