Do spy eyes work like handshakes?

That's the way it seems to work - an early post mentioned you don't have to keep untapped buildings. So if you have any neighbors who aren't participating in the Stonecutter's update, you could save untapped buildings for them. Otherwise, not necessary. I've found I'm making more money because I'm tapping all the cash.

yes. spying also animates the buildings (at least until you leave your town).

Great, now at least I'm sure the spy eyes will gather around the lodge and near buildings anyway.

SPOILER













Yep. And you may want as much money as you can get for this event's end game! Unless you want yet another SPOILER. Wink, wink. Say no more, say no more, know what I mean, nudge nudge.