I've seen the odd villager-into-witch, pig-into-zombie-piglin, or creeper-into-charged-creeper, but never a skeleton horse. Do they come from horses being hit by lightning now, or just at random spots of lightning spawning? Never seen one and want to fight one already.
They are a bit rare, but not as rare of some of the other things you mentioned (charged creeper, namely). They were far more common when introduced and then had spawn chance decreased shortly after if I'm remembering right. Looking at the Wiki, they were formally introduced in 1.9 and it was 1.11 that reduced their spawn chance (20% of what it was before).
They have a chance to spawn when lightning strikes. So you need it to be both raining and for that rain to be a thunder storm. The lightning will sometimes spawn a skeleton horse, which will despawn after a short amount of time. If approached, lighting will strike again (visual only; it can't set stuff on fire) and a few skeleton horses with skeletons on them will spawn.
The last time I saw one was in my newer world in 1.16. A storm had just happened, and a nearby lightning strike must have spawned one. The storm ended, I went outside to a nearby place just outside my village, and suddenly I experienced lightning strike in the middle of the day. Startled me as I was thinking creeper. Turns out there was a skeleton horse in the tree, as ta few of them spawned. I successfully saved one of the horses, and then when I went to recently move it to another village through the nether, it suffocated going through. My fault, I guess.
To this day, in over twelve years of playing though, it is the charged creeper I've yet to naturally see. I'm sure there's some way to encourage it to happen using tridents, but I've never seen one naturally as I've never done that.
Not sure if I've seen a villager become a witch either.
I've definitely seen a pig become a zombified piglin though. Quite recently, actually.
How long does it take a skeleton trap to despawn? How far or close do you need to be?
If you kill only the riders, do all the horses stay passive?
As for obtaining any lightning'd mob, fencing a bunch off in one area and then sitting near it (ideally with a well-placed lightning rod) will help. But I still don't know if lightning bolts create traps or horses + bolts do. I could swear the wiki said it now requires a living horse.