Now that my Oriental poppies have finished flowering, is this a good time to move them, and also to take root cuttings ?
I wouldn't bother- they self-seed like mad (or maybe that's why you want to move them!), or you can save the seed yourself.
Mine are white with purple centres. Seed will only give the basic red ones.
No and No to your questions. Wait until autumn for both tasks, if not next spring. Then lift carefully and take your root cuttings before replanting. HOWEVER, you will find that when you move an oriental poppy, unless you are very thorough you will leave roots behind that will surely regrow and give you a new plant where the old one used to be. That might be an easier method of propagating the plant than fiddling about with root cuttings.
What you can do now is cut the plants back hard to the ground. This will encourage fresh new growth and just maybe the odd flower later on.
Hope this helps.