It effectively can't have that. Quote statuses are practically just a special way of formatting a link to another status in the message text at the moment. There's a proposal iirc to try and create a more genericized version by extending AP, but it's stalled if memory serves me right.
Even so, you're going to have implementations that will happily ignore those choices; as long as the body of an external post is obtainable in some way (right now most info is just in the embed meta tags), someone can make a version of it that ignores permissions.
Any permissions system is, of course a polite suggestion since ActivityPub pushes content to a large number of other peoples' computers in the typical case. The whole network depends on most servers being relatively well-behaved so I don't think that's a barrier.
Perhaps. My comment wasn't based on technical feasibility, but specifying the particular concern which is why Mastodon and several other ActivityPub projects do not support quote-boosts.