Can I see which wishes are booked on my wishlist?

Yes. When someone books one of your wishes, it shows a Booked label, so you know it's taken. You never see who booked it.

In the Wishes app you can filter your wishlist. Tap the filter button and choose All wishes, Available, Booked or Received.

When a gift arrives, mark the wish as received. Open it and choose Mark as received, and it shows a Received label.