I took a TA on Anthem from South Hampton to Cape Liberty at the end of October. Leaving South Hampton was cold and rainy (as expected). I think the high was maybe in the 50s? Waves and wind were so bad we were confined inside for the first two days, with many things cancelled on board (bumper cars, shows), due to all the movement. Personally, I found it to be quite enjoyable. Something strangely scary, yet peaceful.. out in the middle of the ocean like that. There was still plenty to do though. Definitely warmed up a bit coming into NY despite it being early November and how far north the route took us.
As far as the internet goes, we had a terrible connection in our room. Deck 6 JR Suite, mid ship. I dont know if we were in a dead zone on the ship, or if it signal wasn't the strongest out in the middle of the ocean. I wasn't able to FaceTime or stream video when we were on the middle days, but able to check email and Facebook just fine. This was about 3 years ago though.