I'm answering here because I don't think it would fit as a comment but I believe others might have a better answer.
Is each cam, stopper, and carabiner individually tested?
Maybe. It will depend on the manufacturer. Usually a few samples from a batch go through tests. That's why when you hear about recall, they know which batch to recall.
Here is kind of an example (as it's made for the public) of how Petzel run some QA. They make a few pieces, pick some randomly and throw in the test machines.
That means that, somewhere up the stream they made your rope together with another 10 (100, 1000 ?!?) ropes, picked a few of them and put on a stress machine and stretched until they broke. They checked how much they needed to break the rope and give a tick of pass or not pass.
If they all passed, they take the batch as a good and give them the warm and fuzzy seal of approval.
If they don't pass they will take that whole batch aside and check what's wrong.
That's roughly the process.
What do the UIAA markings on my climbing hardware really mean?
The UIAA markings are the minimum requirements for the gear to get the warm fuzzy UIAA seal of approval from the Union Internationale des Associations d'Alpinism (International Climbing and Mountaineering Federation).
The manufacturer would have a number of tests to match the UIAA (in this case) requirements. Other manufacturers might want other seals and have extra/other sets of requirements for that too.
So when they pick some samples and run tests, these tests are to match the UIAA seal.
A few points about QA and certifications
Smaller the number of pieces in a batch, better the QA control. It's easier to have a better quality gear if two in a batch of 10 pass then if 10 in a batch of 100 and so on.
The better brand is not the one the gear doesn't break. It's the one the gear break the least. If 1 piece in 10 breaks is better than 6 in 10. So we just hope we don't get that faulty one. :) (Of course I'm exaggerating here a bit to emphasize my point. Climbing gear has a very high quality standard and I would expect more like 1 in thousands for the crap brand really).