At work we have an industrial washer and we can just about get a king size in there, so I think not in a normal domestic machine unless it is unusually large..
Well I have a 8kg washload machine and I can wash my 4.5 tog kingsize duvet in there but not the 10.5. So it would depend on the size of your washer really, you could try it and see if it fits. Be careful though because it should fit comfortably to be washed properly and trying to cram it in could end up in it getting ripped.
I cleaned a duvet by putting it in a bath, bunging in some washing up liquid, filling with water and then beating and scrunching. when the water was dirty I emptied it, then rinsed through the same way (filling bath) twice, then hung out. It kept shape and might not have been ultra clean but was certainly fresher.
My 3 tog king size fits in my normal washer - if I change to a 4.5 in winter it doesn't really fit so I tend to just buy cheap ones and chuck them when they are dirty.