Mgill

Prices pulled from my recent sailing on Allure (liveblog in signature)
Virgin frozen drinks (like pina coladas): $7
Specialty coffees (including lattes): $4-5 
Fresh squeezed OJ: $5
Smoothies (if you are on a ship that makes them): $8.25
Water: ???? prob $2. Unknown because the bartenders never bothered to key it into the POS once they saw you have a beverage package. 

Specialty coffee is in line with starbucks prices on land (but the actual Starbucks kiosks on the ship are not included in the package), $4-6. I believe the virgin coctails run about $8 from other posts I've seen here somewhere. Water & Soda I'm less certain of, I think at least $2 for a bottle of water is a safe assumption for doing the math, soda is a little different since with the package you'll get a reusable cup for the fountain, but you do get access to cans or dispensed soda at bars too. I expect they'd charge for a cup of soda like a restaurant would, $3-4, but can't be certain. You also get access to the fresh squeezed juice at breakfast in the MDR, which I believe was listed at $4.50 on Enchantment last year.
For us, the math works out, because we'll down 2-3 of the coffee per day on vacation, juice at breakfast, and plenty of soda and mocktails along the way.

No.  If you have the UDP you should be able to go to Chops (or whatever specialty restaurant is open on embark day).  I have heard of a few singular instances where people with an UDP were turned away but that was early in the days of the new and improved UDP and this have never happened to me.  I think each ship handles it differently though.
Are you in a full suite (Grand Suite or higher) ?  I always have the suite concierge make a reservation for us on embark day just to make sure that they are booked expecting us.  You also should receive the “regular” lunch menu as opposed to the more limited Key lunch menu with the UDP.

I had the Unlimited Dining on Navigator last month, yes if a restaurant is open you can dine there.  We stayed onboard in Nassau and had lunch at Jamie's Italian.