Where are registers and what do they look like?
I still don’t understand what a register physically is... Where on a
  processor is it? What do they look like? How big are they?


Physically it is made from the same structures that make up all the other logic in the processor. Depending on the implementation that could be transistors created on a silicon wafer, or discrete transistors, or vacuum tubes, neon lamps, electromechanical relays etc.. 

A modern CPU may have billions of tiny transistors and multiple cores which make it tricky to spot individual registers and their relationship to other parts. However the same principles apply to simpler CPUs such as the Intel 8080A, shown below.

      

The area marked '16 bit register array' contains the flip flops that make up the BC, DE, HL, SP and PC registers. They are located close to the ALU and Address buffer for fast communication with these subsections. A register can be connected to the internal 8 bit bus via tri-state buffers for access to the accumulator, ALU and external data bus, or to the address buffer for addressing external memory.         

The chip dimensions are 4.28 mm x 4.18 mm, so the entire register array takes up an area of ~1.2 mm2.

After removing the metallization layer and zooming in we see the fine structure of the registers. From this it should be possible to identify individual transistors and derive the equivalent electronic circuit of each register.



Of course this pattern is specific to the structure and layout of this particular chip - another CPU could look quite different. But what doesn't change is the functionality. No matter how it is implemented physically, a register still does the same job logically.