Haven on Gem - I have booked and sailed on several Gem balconies in the Caribbean and loved the experience. On one booked balcony, we won the bidding system and received a discounted Haven 2-Bedroom.
Don't get me wrong, the Haven is VERY nice, but on the older Jewel Class Ships, the Gem does not of all the bells & whistles of the new ships. There is a small courtyard with a tiny essentially a wading pool with a retractable roof that can be rolled open/shut depending on the weather. The private sun deck was up one flight of stairs and was virtually empty for our 10-day cruise. The second bedroom was very small and had two singles beds, very stiff beds (really not Haven quality, but available). The master bedroom is very nice, there is a beautiful private deck, small living room, small table & chairs for eating in your cabin, one giant luxurious bathroom and a second balcony side bathroom.
There is no bar or restaurant in the Gem Haven. Breakfast and lunch was served in Cagneys exclusively for the Haven & Suite guests. The butler could be utilize to bring bar drinks to you in the Haven (to the courtyard, sun deck or your room) from any of the "regular" bars. Additionally, dinner or any food could be brought to you by the Butler, as well as any little num-num snacks during the day. The concierge will get you any priority seating at shows, restaurants, excursions, as well as embarkation / disembarkation.
To repeat, its a VERY nice experience and you will want for nothing, but not as nice as the newer Havens.
In your particular case, if you booked two balcony rooms, all guest would be entitled to all five offers. However, your 6-year old obviously does need the liquor plan. You are then paying for two cabins.
With that in mind, if you booked the more expensive Haven, first two guests receive all five offers. As of right now, guests 3 & 4 can sail in the Haven for "free" - no fare but you pay the port taxes and gratuities only.
In summation, you may want to research pricing for two balcony rooms versus one Haven room. Do a mock booking on both options for pricing. Pick your rooms in this drill - NCL will advertise a Balcony as low as $1,049, but this is a guaranteed room and comes with ZERO perks. To get the 5 perks in a balcony you need to book each cabin somewhere between $1,300 to $1,600 each. Single Haven 2-bedroom sells for $3,900.
Here is what I see in TAKE ALL 5 issues on a seven day Boston to Bermuda. With the Haven you would need to separately purchase the number of specialty dining nights for two guests. NCL will give guests #1 and #2 three specialty dining in the PICK 5. This again assumes the 6-year old would be happy in a steakhouse, French restaurant Hibachi dinner experience, etc. as opposed to dining in one of the two the main dining rooms. Drink package - depends on the volume of drinks consumed. If alcohol usage would be considerable, then consider purchasing one drink package for the third over-21 guest. If one of the three adults is not a heavy drinker, rather than buying a Beverage Package and spending $99 per day on the package, just purchase individual drinks (to make the package economical, your "weakest" drinker would need to consume 8-10 drinks each day, including the port days in Bermuda where you will probably be off ship most of the day). Hint: Make sure the two "stronger" drinkers are listed a Guest #1 and Guest #2 if booking the Haven. The $50 per port excursion benefit is per cabin, so 2 credits for 2 Balcony cabins verses one credit in the Haven. Cannot bank or stack the credits - if don't use it any day, you lose it. Again economics depends on the number of NCL sponsored excursions taken. If you use an outside vendor or go on your own, the $50 credit is worthless. No need to worry about getting back to the ship on port day 1 or 2. Day three is obviously critical. If Wifi access is critical, the TAKE 5 usually gives 250 minutes per cabin. Three days in Bermuda - I'm relatively ceratin that the 2 twenty-somethings can easily locate free Wifi in shopping areas / restaurants etc..
My personal opinion - If I were looking to splurge, I would go for the Haven.
Either way, it sounds like a great vacation - Enjoy!
Several years ago we did an August NY-Bermuda on the Dawn. It was VERY hot & humid for our cruise. YMMV