My personal opinions on the subjects are;
If the post weld heat treatment is required, it belongs in the WPS. Whether it needs to be qualified is a subject that should be addressed by the applicable code. AWS D1.1 is intended for structural applications where both the carbon content and the carbon equivalencies of the steels are limited. The post weld heat treatments most often encountered are for the purpose of stress relief, but even for that purpose, stress relief is not a frequent operation for structural applications. Even the hydrogen bake-out is not a normal production operation in structural applications.
The same cannot be said of weldments that are used as pressure containment. ASME construction codes require post weld heat treatment for many applications and alloys. They are more definitive and place requirements on what, when, and how a material is to be heat treated. Section IX provides fairly explicit definitions on different types of post weld heat treatment when it comes to qualifying the welding procedure.
AWS B2.1 can apply when AWS D1.1 isn't applicable and it is closely aligned with ASME when it comes to essential and nonessential welding variables, including post weld heat treating operations. Welding procedures that include PWHT have to be qualified using the same HT procedures as will be used in production. As such, B2.1 requires the WPS to be qualified with the appropriate PWHT conditions as follows: no PWHT, PWHT below the lower transformation temperature (which in my opinion would include hydrogen bake-out as well as stress relief at temperatures below 1330 degrees F), PWHT within the transformation temperature range (slightly above 1330 degrees F), PWHT above the transformation temperature, etc. There are additional requirements, but I think the general approach is very similar to Section IX.
As for flame straightening, it isn't a welding process, but if it is a controlled process, which it should be, there should be a procedure for it as well. Many of the states have specific requirements and limitations on how flame straightening is performed when the weldment is intended for a bridge.
Best regards – Al