I guess I could qualify myself as a 'Bermudaphile' only have been going there for 30+ years! You questions are very good, and I'd like to help with a few answers. First, I know that 'open dining' has it's advantage/disadvantage. I would definately let the maitre'd know IN ADVANCE of your needs, and request a time to eat. I think they'll be able accomodate your reuqest.
As far as parties, I'm not familiar with all the details, but having been on several cruises, I tend to stay away from 'party time' just because....I choose not to participate in them!
As far as your father's needs, you may want to call the Bda. Dept. of Tourism in NYC and inquire about a special van or vehicle that accomodate you and your party. Normally the taxi's are pretty roomy, but with your needs, you may want to tell then what specific needs you have for transportation, and they can point you in the right direction. I know of one or two specific companies who deal with such issues, and I hope the Dept. can be of help.
Bermuda is a 'low pressure' island, and the people are very hospitible and friendly. If they say 'good morning' to you, they expect you to return the compliment. If you want a relaxing day, the beaches on South Shore are a great wayto enjoy the water, air and your daughter.