What is the best time for grafting deciduous fruit trees like apple, pear and peach? I have been doing well with the grafting just before or after bud break, but can it be done in the growing season or mid winter?

When after summer heat rain returns in August that's the time for T-budding here.
Appricot could be T-budded these days.

If you are T-budding, June is the favorite month. You want the rootstocks growing vigorously, so the bark will slip. Late summer budding also works, if the stocks put on another growth flush. Chip budding works nearly any time as long as the scions won't be frozen for much of the time after grafting but before they heal.

Can you graft brown, or dormant scions onto green branches?

What about vice versa? Green scions onto brown branches?

I have never tried, but the idea behind the winter wood on growing stock is that the winter wood does not need much sap to stay alive and that gives the graft more time to connect and start supplying sap to the scion.

To do it the other way around is going to leave the scion in high demand for sap with no supply--I do not think it will work.