As I understand it ActivityPub uses a combination of push notifications at time of publishing and pull notifications at time of subscription/query for objects?

It's a mix of pushing and pulling. When something happens, the server pushes a notification ("activity") to other servers. But recipients often need to pull additional data, such as user profiles or related posts.

Duration of caching is set by the instance admin I take it?

Yes, and it also depends on the software. Some applications may keep cached objects forever and only prune cached media (because objects don't require much space).

Regarding Authorship, if there wasn't an issue then ATProtocol devs wouldn't have made it the cornerstone feature of their network

Moving in ActivityPub world is difficult because authorship is tied to a specific server. We can solve this problem by using cryptographic identities and signing everything, like ATProto and Nostr do.

I'd like to know how delete requests propagate, when the "Object" is deleted does a request to clear cache go out to all federating instances?

Deletes and edits are usually sent to followers of a user or a community. Delivering them to all known instances is not practical.