Is there any untamed physics in the CPU? In a x86/amd64 CPU could I find an instruction or a combination of instructions that leak inside the system? Could some other physics properties (not accounted by the design) be measured or interacted with?
 A: As with any universal Turing machine, there is no general way to predict the behavior of most software. That includes the behavior of your "CPU + memory" system. One specific behavior that you cannot predict is if a given input will result in an output or if the machine will keep computing, or running, forever. Of course, you can predict that for simple software examples but not in general 
(see halting problem). 
I am not sure what other kinds of specific behaviors you have in mind, but I'll be glad to try to answer it if you can make your question more specific.
A: yes, it is possible but unlikely. Manufacturers do the best to deliver hardwares matching their specifications , using as many errors corrections algorithms that needed for this purpose.
Some people look for expected quantum signals at the i/o devices level. For example, a physicist claims to use an ordinary camera to get quantum random numbers, while others try to count the occurences of some crc corrections. I am not convinced that is what is expected.
