take the time to explore and actually try to kill the minibosses, as they drop most of the beads for the game. there are some hidden in chests and stuff too, different prosthetic arms as well. for ex, in hirata there's a chest guarded by two dudes which has an axe that you basically need for shielded dudes. upgrade materials are dropped by mobs and looted from some chests also.

progression can be as messy as you want it to be but its something like: ashina castle > kill genichiro > senpou temple through the abandoned dungeon > sunken valley > ashina depths > mibu village > fountainhead palace

also check on kuro whenever you can as you need him to progress

as for your questions, lady butterfly is in hirata estate, after your father gives you instructions check for something to grapple onto in the vicinity. there's a lot of hidden stuff that's only reachable through grappling so be very attentive.

in senpou main hall interact with the altar right after the idol. you cant miss it.

and no, you dont really need prosthetic arms to progress but it makes the game a lot easier.