Catching spies seems to be a "passive" mission, that occurs automatically with no input from the player (or the AI.) Thus, a message will pop up saying "you've caught a spy!" Even though you did nothing specific. Counterespionage raises the cost of active missions, and presumably the listed "passive" missions, but whether it makes catching spies harder is one for the code breakers. I would doubt it.
Traditionally, wonders have been immune to destruction, so just guessing, but I doubt you can destroy them with a spy.
As for the security buildings, they can be expensive to destroy, and so may not show up on the menu if you don't have enough espionage points. I've seen the cost well over 2000 points to destroy the security bureau, with a chance of success below 60%. And that was in a country where I was outspending them on espionage.
You're right, more info on spies would be nice, but there are lots of "undocumented features" in this game....