A dense fibrous root system and habit of shallow rooting make them among the easiest of shrubs to move. You can assume the roots extend as far as the widest branches but you can take a somewhat smaller rootball safely if necessary for a manageable weight. Drag on cardboard or tarp, don't try to lift and have the weight of soil on roots damage those roots.

You can move them any time they are not in active growth (immediately after flowering) or the ground is not frozen - but I'm not familiar with your Fall/Winter temps. Best results may follow when you observe the timing proposed for planting new plants in your gardening zone...are you anticipating weather that would freeze ground any time soon?

Don't cut it back. If you have the opportunity now to add more organic material (compost) to the bed do that. Be careful not to plant any deeper than originally growing (if you've spaded through the area this may mean planting a little high to allow for settling), and apply a mulch over the root zone after transplanting to help conserve moisture, always important but especially so now to help prevent frost heave over winter.

Your azaleas set their buds for next Springs flowers in late summer - there is nothing you can do at this point to encourage more buds.