Would the form factor/board layout of the Particle Photon be described as the “Feather” layout? If not, is it some kind of other standard breakout board form factor?
The Photon does not follow any standard. The upcoming Mesh boards will be feather compatible.
Particle provides the Gerber files for the Photon board though, right? So if I wanted to build a similar board, or modify the existing board using the same board layout I could get all of the dimensions from the Gerber files?
Have you seen this (with regards to the form factor part of the question)?
Also I would go directly to the Eagle files if you want to build a modified version of the Photon itself. The schematic and board files for Eagle are there, and that’s what the Gerbers are generated from.