How long should I stir French press before "pressing"? What are the pros/cons and effects of stirring, either "too much" or "too short", in a French press?  My rule of thumb for steeping is to let the coffee cool down to a drinkable temperature, then serve immediately.

I was going with the thinking that in whatever chemical reaction is occurring during the brew, I would want to accelerate it with maximum stirring.  Particularly when maximizing caffeine?

The video here:

https://en.wikipedia.org/wiki/French_press#Preparation

shows a quick stir only, however.  They mention a range of brewing times.

(side question:  the only reason not to stir with a metal spoon is to prevent breaking the beaker?)
Based on my experience, I can say that stirring does not affect the brewing process in a French press a lot. The initial stirring is just for homogenously distributing the grounds in the water. Then, the rest of brewing happens itself in the water by chemical extraction itself and the main factors are time and temperature.

According to your question, I can say that you may be over extracting if you are waiting for the water to cool down.

One quick recipe and other related discussion are mentioned here.