Your family should really enjoy the Explorer.
To answer some of your questions:
1. Yes, you can pre-register online once you have made your final payment. You will just need your booking number.
2. Here is site that has many of the RC menus: www.cruiseclues.com
3. Miniature golf is always open and there will be scheduled times for ice skating and rock climbing. I know that golf and ice skating are on a first come-first serve basis, but not sure about rock climbing (I beleive that is something you sign up for).
4. I have found noise level in the promenade staterooms a non-issue, other than during the occassional scheduled parades and even then it is not overly invasive.
5. Food is included on Labadee as are some of the activities. Water toys/snorkeling gear, kayaks, jet skis, etc. incur an additional fee. Here is a link to the information we have compiled on our sister site Port Reviews: http://www.caribbeanportreviews.com/Labadee.htm
6. Tipping is about $10 per day per passenger and you will be given the option of automatically adding it to your onboard account.
Hope this helps-