Can I kick off a manual search for all missing episodes or do I have to target a specific show/season/episode?
Yes, on the Wanted: Missing page there is an option, beware that it will use a large number of API requests and might take a long time to finish, especially if you have limited API calls or your indexer tells Sonarr to limit its API calls.
Does Sonarr “know” when it hasn’t been able to catch up on the previously loaded feed?
Offhand I’m not sure if it tracks it anywhere beyond a log message, but it does log a Warning (pretty sure its a Warning), when it can’t catch up.