As far as I can tell (not that I've tested these properly or anything, just speaking from experience), the Pokemon Professor event always appears to occur first in an episode, followed by the farmers asking for money, and from there it's almost always up in the air.
When I was Eevee hunting, I was relying on that same event (the one where a single Eevee is seen alongside other Pokemon, not a swarm), which seemed to occur about once every two years (or at least, it took me about 10 years of playing that episode to get 5 Eeveelutions). It may be that an Eevee will only show up if you have the MC, and said MC does not already have an Eevee (but does have one of its evolutions) - certainly, I never managed to get two Eevees at once, despite spending about three years trying to get an Umbreon, and have not got an Eevee event since grabbing all of my Eeveelutions. On the other hand, it did not seem to be pre-determined - the first one I got (in Mori Motonari's episode) came along after only about 5 months, whereas I had to wait much longer (in Maeda Keiji's episode) for my next one. I'd be willing to bet that it's just random, but has only a reasonable chance of occurring.
Oddly, they all seemed to appear in Pugilis for me, rather than Cragspur, irrespective of the castle that my MC was based in.
Hideyoshi's event still isn't showing up... I guess they've probably forgotten about it. Still, if the worst comes to the worst, there's always that password that unlocks it anyway.