(e.g. how do you represent where a stiffener layer is placed?)
No need to overcomplicate it. Just make a Gerber layer, draw where you want a stiffener or an adhesive and include the dimension layer, and label the file with something that indicates the purpose. I suggest "[PCB name] [rev number] - Top/Bottom Stiffener Layer" or "Top/Bottom Adhesive Layer." Just make up a unique extension. Or not. I don't think there's any manufacturer out there can't figure it out. The old "standard" extensions are pretty much obsolete, and if the manufacturer wants you to use a specific extension in the future, they will tell you.
And attach a note as to how thick of a stiffener/adhesive, and what you want it to stick to. Then let then come back to you with options... They have certain materials on hand which they prefer to work with due to sourcing/cost/familiarity. They'll be able to suggest 3M xxxx or what not that you can look up the specs on.
It's nice to know these are options. But OP, there's no need to stress about it.
Many of the major manufactures' quote system will show you at least some of these options, particularly where it comes to the board material and thicknesses available.
IME, the minimum parameters are all the same, regarding minimum hole size and trace width without incurring extra cost, etc. Pretty much the same thing as ordering a rigid PCB, except prototypes cost more. I guess there isn't enough demand to make pooled services as efficient. In volume, the cost isn't necessarily much different.
For a cheap proto, check out ITEAD. For their cheap proto service, they do have specific extensions they request for the standard Gerber layers, but it's all detailed on the site.