There seems to be some confusion about your intent here. I read it as you don't intend to inspect for a week? That seems fine to me. However, if you have somehow restricted your bees to the hive, I agree with @Rodderick and @JeffH.
If you started with a nucleus in March, at the beginning of the nectar flow season, you may get a harvest. However, starting in mid-June it is going to going to be a challenge for the bees just to lay away enough stores for next winter.
Err, yes...? Sorry your question was a bit confusing to me. The answer to the question I think you are asking, is if you have 2 brood boxes, they will put both brood and honey/pollen in the brood boxes. They will use this during a dearth, but if you left a honey super on over winter, they would also use that. I would not recommend leaving honey super above a queen excluder over winter in your climate.
In Washington state, you will certainly need 2 deeps for brood to help them survive over winter, and you may need to feed them as well. I would get both brood boxes filled with bees and stores before you put any super on the hive.
Yes, but make sure they really are full. That means:
1. All frames have fully drawn comb over most of the frame, and
2. The comb is 80% filled with food or brood, and
3. Every frame is covered with bees.
In fact that rule applies to adding any box to a hive. When you put the nucleus into the hive, it should be in one box of 8 or 10 frames (depending on hive size). When that box fits the above description, you can add a second brood box. When the second brood box fits the description, you can add a super. Etc.