Do you think they will move the nectar from the deep to the shallows?
Yes they will, but it will take a bit for them to draw the foundation first.
I would never recommend using an excluder on a colony trying to draw out foundation. This is the biggest mistake I see for newcomers. Let the queen lay where she may, the more brood that's reared the more workers you have to put to the task of drawing comb. After the first year, if you insist on restricting the queen, turn a queen excluder 90 degrees so there is a gap along the walls for unrestricted traffic. The workers will move up easier, and the queen will not venture that close to the exterior of the broodnest to find the open gap.
Can I removed any of the frames of honey from the Deep? and if I do will the bees have time to build up enough honey for winter?
Sure, but it's hard to say not knowing where you live. We could give you a more proper response if we knew your geographic location.