In this case, person B should have purchased a more expensive and better quality security system. Yes, we native speakers certainly do use the expression!

The expression, 'when you go cheap on ...' refers to yet another expression. That expression being, 'You get what you pay for'. Both expressions assume that when someone pays for an item, or a service, the more expensive the item, or service, the more reliable, or higher in quality, it is. Most would agree that this is considered a "Maxim" of life and is an accurate rule of thumb or generalization. This maxim might not always apply, but it holds true for many things.

The sentence simply implies that because he bought himself a cheap security system, the burglars were able to break in and steal from him. Another example might be: A: Oh no, I've just torn the backside of my jeans. B: That will teach you to go cheap on jeans. It's not something I would say but I have heard younger people using it.

"choose a cheap option" It sounds natural. There are probably many occasions where you can improvise a phrase consisting of "go + adjective". If you go to McDonalds and ask for a Big Mac meal, they may ask you if you want to "go large" i.e. choose the large version of the same meal.