If I capture a bunch of workshops on the map while playing on a private server, will they all remain captured the next time I log back in to the same private server?
No, they will get erased but not immediately. Sometimes the game can crash for whatever reason and if you log back into the game right away and go to a private server, the world state should be saved. This means all of your workshops will stay the way they were. This normally lasts for about 15 mins before resetting and after this the server shuts down and all progress is wiped from the game.