1. Both Firefish and Mastodon get their feed the same way: ActivityPub server push messages. Each server sees only that content which is federated to them, or which they explicitly fetch from origin server (due to reference in another toot, typically).
2. Search is implemented as an ElasticSearch (or similar) index over the public toots. Mastodon has had this index before, but not show toots from it to your search unless they were your own, or you were mentioned, or had engaged with the toot before. That's changing now.
3. Firefish antennae are prebuilt, recurring search subscriptions. Mastodon's closest equivalent is subscribed hashtags. Antennae are conceptually similar, but allow subscribing to a more complex query than a single hashtag.