Cannot establish serial communication to ESP8266
After scouring the internet for many hours, and staring at my board and schematics for many more, I've finally resolved this. Hopefully this will be of use to someone someday.

There were multiple issues. 

Two of them were both critical to comms over UART:


MTDO (GPIO15) must be LOW during boot, for both booting from flash or UART. I had read online during design that it had an internal PDR and therefore I wouldn't have to worry about it. Despite that, as you can see I even have an external PDR (R483 -- 100k). I eventually was able to find a source from Espressif stating that MTDO actually has a weak 37k PUR. I replaced R483 with a 5k PDR. 
The datasheet for the ESP8266EX says the minimum valid crystal frequency is 24 MHz, which is what I was using. This does not appear to be the case. Old versions of the datasheet say 26 MHz. I replaced Y1 with a 26 MHz crystal. May also be due to my crystal's stability being out-of-spec (50 ppm v. 15 ppm).


After implementing both of these changes, UART now works as expected. Uploaded programs still behave unpredictably. GPIO is unstable. This leads to the single moderate error:


Flash chip (256 kB) is too small. Replaced with 1 MB.


Uploaded programs now behave as expected. Wireless connectivity was the last issue to address. It works, but is very sporadic and seems to brown out the MCU, causing constant reboots. Leading to the single mild error:


10-uF electrolytic added to input side of regulator. 


After these changes, everything works exactly as expected.