If you don't want your PCB copied, then don't send it to China. Also, PCB is only half of the design, they don't have the component values or firmware\software. However, if someone really wants to reverse engineer your design, they will find a way. It's not hard for a customer in any country to order your design (if you're selling it) and copy it verbatim. Even scratching off component values won't help, as you can de-encapsulate ICs and get a good idea of what the chip is.
There isn't a good way to evaluate a PCB house (or any business) before purchasing. But you can inspect the PCBs post production. Most PCB houses will give you a cross section of the board, which can be inspected under a microscope. This will allow you to check the layers and copper weight. Usually PCB manufacturers offer (now most are free) an electric continuity check, and they check the continuity of all the traces. Most board houses will do an excellent job; they want repeat business. If you find a mistake (and it was their mistake) ask them to correct it, many will.
I have had a few issues with board houses, but it doesn't matter the country of origin, sometimes mistakes slip through. If there are enough problems, I switch manufacturers. No manufacturing process is going to be 100%, but it's reasonable to expect higher than 98%.