Here are some answers to your questions:
1. The 2 hotels are across the street from each other. It's a wide street, but totally navigable, probably even post-party.
2. Yes, Video is ok
3. No. The Nieuw Amsterdam is brand new, but if it's similar to the Eurodam (almost brand new) there will be 2 restaurants with an upcharge. Details should be on the HAL site, but usually the top-tier restaurant is The Pinnacle Grill, which costs $25 per person, and offers exquisite food and service. The other upcharge restaurant will be an Asian theme, and will be less $$/person, and will also be absolutely delightful. I'd be willing to have all my dinners in those 2 restaurants EXCEPT that it takes too much time away from the music (or from catching up on sleep). If you enjoy the late night music, time will be your most valuable commodity. Because of that, many many cruisers even skip the main dining room, opting instead to eat in the Lido Buffet, which is always good food, and is much faster.
4. No. Internet access is expensive and slow (it's via satellite). See HAL site for details. Seems like every year there is more widespread wireless access. Last week, we got excellent signal in our cabin, but I know that's not true for all cabins.
5. Scheduled Pro-Am Jams always have a Jam Leader who manages signups, flow, etc. There will also be lots of less structured jams throughout the week, both in the Crow's Nest (where the Pro-am Jams take place), and anywhere there is a piano. What do you play?
Don't hesitate to ask more questions, and check out earlier posts, as this category has collected a lot of useful info over the cruises..