Like it's been said above, browning the meat is just so the sugars caramelise and add to the flavour. I'd definately do it until there's a residue on the pan itself and possibly even little dark flecks on the meat. At that point use some wine or some stock or something to deglaze the pan and get all that flavour into the sauce.
Regarding the wok, it shouldn't really matter. I know woks are meant to have high heat in the center with cooler areas up the sides to stack cooked food, but it should be fine (if incorrect) for quickly cooking other things. The main issue may well just be the quality of the pan. If it's a £5.99 job from a supermarket, you could probably burn water in it half the time.