What are the reasons for non-uniform temperature distribution across gear teeth face-width?

There are different amounts of heat being generated.

When does the book come out?

Op
Normally contact stress and bending stress. Also the inherent gear teeth errors such as lead , (tooth parallelism), involute error, and total composite error, and tooth to to error.
So that fine thin layer of lubrication prevent frition which causing heat. Incorrect lubrication and gears will fail very quickly.
Heat and friction will be generated.
It is well know more precise the gear and surface finish. The longer gear cycles.

SMKRe - you ask some fairly specific and perceptive questions. You should try answering them yourself. I think you could.