Congratulations on the upcoming anniversary! And also for your first cruise!
Here are some brief, high-level answers to your questions
1. Nassau, in my opinion is a fine stop. If you've never been, check it out.
2. Absolutely, especially for such an occasion as your 20th Anniversary
3. Must-do on the Ship: Slow down and take some time to just explore ; on Castaway: Serenity Bay (Adults Only beach)
And now if you're interested here are some much more wordy answers:
1. Nassau is not without its faults, for sure. But I think the biggest thing against it, especially for repeat Disney Cruisers, is most itineraries stop there, so after the 2nd visit we tend to see this as a reason to enjoy a partially empty ship rather than a destination. There are some worthwhile attractions here, such as the Atlantis Resort, Blue Lagoon Island (for the beach and/or dolphin/sea lion activities), a guided island tour, etc. Venturing off on your own, however, you will find that the locals can be excessively pushy salespeople especially in and around the Straw Market. That being said, even if you aren't interested in any of the excursions, nor exploring by foot, it's worthwhile for a first visit to get off the ship. Once you leave the pier you will find yourself in the Cruise Terminal (Festival Place) where you can enjoy perusing some stalls with local foods, crafts, and souvenirs within the comforts of a gated building (as opposed to the Straw Market where you will find the pushy sales-folk). Here, you can also get your Passport stamped, if you are so inclined.
2. Palo is a luxury dining experience, and will beat out the menu of any of the rotational dining restaurants or other dining options, hands down. It is also, affordable at $40/pp. Off the ship you will not easily find a restaurant where you get that level of service, with that amount of courses, for anywhere near that price. Furthermore, you are on a 4-night cruise, which means you will be repeating one of your 3 rotational dining restaurants. If you can plan your Palo visit out to align with your repeat venue, you can rest easy without a fear of missing out on something. (But even if you do miss a dining venue, Palo is a superior experience and it's ok)
3. As for must-do's on the ship, it depends on which ship! Based on your itinerary I'm relatively confident you will be on the Disney Dream. I'll reiterate my recommendation to explore the ship. There is a ton of detail and hidden bits of magic you will find throughout. There are several tours you can attend that will point out some of this level of detail (all free, of course). I realize everyone vacations differently, and so does DCL. Check your daily navigators (The Disney Cruise Line Blog website is a great resource to look over previous Navigators to get a general idea) and do whatever appeals to you! My wife and I love so much that every day is jam packed for us - from Mixology classes, to game shows, variety acts, movies, swimming, dining, etc. we never feel bored! If you don't mind splurging, the Spa is a personal must-do for myself, but is an upcharge service.
Castaway Cay is our favorite stop, hands down, every time. The 2 family beaches are great. I, especially, love Pelican Plunge, a floating structure between the 2 family beaches that features 2 water slides. The snorkel lagoon is fun if you enjoy swimming. There are buoys throughout that indicate where you can find some sunken features (derelict ship, mickey statue, etc.) that fish love to hang around, but it is also a pretty far swim and I see a lot of people get tired out by it. Worth checking out if you've never snorkeled before. There is a 5k if you're into that sort of thing that takes you on a decent path around the island. There are also several excursions you can do here (Parasailing, Jet Skis, Etc.) - I personally favor the Kayaking tour as you get to walk around parts of the island that are otherwise inaccessible, as well as Kayak through the Mangroves for a unique experience. But the shining gem here, as I mentioned above, is the adults only beach Serenity Bay. It is protected by a sandbar so the water here is usually very calm, still, and shallow. You can usually walk out to the sandbar without ever going more than waist deep. Anecdotally, I also prefer the dining option out here (Cookies Too) as it is less crowded with the same menu.
I'm sure you'll have plenty more questions as the days draw near. We're here to answer them for you. Have fun!