Embedded processors gate length
New Intel and AMD processor feature processors with 14-22nm gate
  lengths, and this information is widely available.


The reason this is so highly publicized is probably historic more than anything, relating to Moore's law. There probably isn't a company more ingrained in the computer market than Intel (Gordon Moore's company). They have been pushing the process envelope for years. Regardless of how we got here, it's become a race to the bottom between the foundries to see who can get to the smallest process node first. Whoever that is, is going to have a big payday, making the bleeding edge SOCs of that time.


  However, when it comes to microprocessors (PIC, ATMEGA) I have not
  been able to find this information. Is it "classified" ?


It's not that it's classified, it's more than no one cares. Wind the block back a few years and the choice between AMD and Intel for your computer when you bought it was a big deal. No one cares whether there is a PIC or ATmega in their toaster oven, let alone what process node it was made in. The designers don't care either. My company uses PIC almost exclusively. Process node was not a variable when that decision was made. It was all based on cost, tools, availability, and support. For that reason, Atmel and Microchip are probably still making those lines of products in a micron scale fab because it's paid for.


  As to ARM processors, since ARM is an architecture, who defines the
  technology to be used when manufacturing say an Cortex-A8?


ARM only licenses their core, they have no say in what process node is used.


  I used this example because the Beaglebone Black uses a ARM Cortex-M8
  manufactured by Texas Instruments, so which of these entities sets the
  technology? Again, unable to find information related to this.


Again, you're not going to find that information because the people buying those chips don't care. TI may tell you if you ask them directly (maybe only after you buy a few thousand chips), but they have nothing to gain from publishing it.