It used to be every 30 that you got a rare. I think they may have changed it to every 60. I'm not sure since I don't pull from there, so if a mod could help, its much appreciated. But there is no particular order in which you get a ninja. Every 120 pulls you get Rasa/Gengetsu (unless they changed this as well)

They haven't changed it to every 60.

Seems like every draw 10 I'm getting a rare in Edo on *t.

The most I gone without a Rare when I draw 20 and get a Rare at 20.

It'll most likely be Ginkaku, Mu, or Kabuto.

I got the ninja in this order

Ginkaku Kinkaku Ginkaku Rasa Kinkaku Mu Mu Kabuto Rasa Kinkaku (No Rasa

20-30 scrolls on average for a rare. 110-120 for a super

The counter in edo tensei is 40, but the chance per single draw to drop one is around 3% so is very likely you get a rare by lucky roll that reset the counter during the first 3 draw by ten you do.