We stayed at BSV late June/early July of this year and really loved it! Don't know about requesting specific units, but the unit we had (#12 I think) was great. Just keep in mind that only the bedrooms are air conditioned. There was a flat screen TV in the living room. As for umbrellas, there were beach loungers provided, but no umbrellas. We thought about going to the store and buying one, but we found that we didn't need one since the covered deck is so close to the water that we spent much of our time there when we weren't walking the beach or swimming. The breeze and shade was nice there and it was better than sitting under an umbrella. Plus you're so close that you could flip a coin to the water!
Ed