The current cruise shouldn't be affected at all. The next cruise may want to get out to sea a bit earlier to come around the back side of Lee to get to Bermuda
For clarity, the app is the only way to grab a time slot and then come back later to finish.
But if you prefer, you can finish the check-in on the website even after using the app to grab a time slot.
I'd bet you'd get info by Tuesday. By then they'll have a better idea of what Lee will do that far north. But wouldn't be surprised if you see a change. The storm is moving so slow that by Friday is still may be south of Bermuda
Waves are probably why they changed itineraries so soon. Waves won't threaten the ship, but more people than usual would be uncomfortable on board and would make those port stops more difficult further south.
Wearing what you want will make you fit in since there will be all manners of attire in the MDR on formal night. Wear a tux/evening gown? You'll fit in. Wear shorts and a t-shirt? You'll fit in.
That happened to us in March. 3:30am, captain came on to give instruction to passengers to stay away from the helipad and keep blinds closed/lights off in the front of the ship.
Problem was the announcement lasted what felt like 10 minutes.
5Mbps is more than enough for typical onboard internet use.
That upload isn't good though if you feel compelled to share every minute of your trip on social media.