Many owners will take mid week bookings - I for one certainly do. As for discounts for 3 weeks, this depends on the owners. I do offer a small discount for booking of 3 weeks and over, but some owners do not.
As you require 3 weeks it would be best to book in advance. As long as you have a rough idea of your dates, you can probably reserve a villa now. Also you may find that you will be able to book at slightly cheaper rates than waiting until next May.