No
There are many ways for classified information to be disclosed that do not involve a violation of the law. For example, classified information could be left in a filing cabinet which is sold at auction to a national broadcaster - as happened in Australia. While careless and embarrassing, it’s not illegal for the government to sell its secrets on the open market.
Government has (or should have) many ways of keeping track of where their information is and who has access to it. The government has to demonstrate not that other ways are hard, time-consuming or expensive but that they literally do not exist.
Bullshit. Many, many things are over-classified because the person doing the classification has every incentive to be overcautious with every piece of information. “Should the officer’s club menu be classified? Well, they are having curry on Friday and that could easily hide poison so ... better to be safe than sorry.” Example deliberately taken to extremes for comedic effect - but you get the idea. I’ve held Secret clearance in the past and the amount of stuff I’ve seen marked “Secret” or, even more so, “Classified” that leaves you asking “why” is mind-blowing.