Responsiveness is at least on par with the other Pros from 2015-16...
Probably a little better but I'm not really sure as the older Pros were also very responsive.
I actually don't have an Apple Pencil so I can't comment on that.
And as for speed, I do actually notice a difference when scrolling on webpages. It's a minor difference but an improvement nonetheless... it's like where sometimes the webpage might stop scrolling for a split second and then jump, now that doesn't happen at all. Probably has to do with the 120Hz refresh rate and/or A10X... it's just smoother all around. (Not sure actually if that falls under responsiveness or speed but it is noticeable either way).
Haven't played games or anything on it yet, but I did watch an episode of a show from iTunes and that was smooth and sound was crystal clear and loud. Loaded all my apps and movies, shows, ebooks etc... and downloads were fast over 5GHz wifi with no problems whatsoever. Tested out the Kindle app, Evernote, Safari, Mail and everything was smooth and fast.
Battery life seems a little bit better than my 3or 4 month old 9.7" Pro and my year and a half old 12.9... but of course that may just be cause it's brand new. I really put it through its paces this past 24 hours and it was at 59% when I finally decided to charge it again. Charging seems pretty fast too. When I checked how much I had used the iPad with actual screen-on time in those 24Hours it was around 4 hours and some minutes and it was at 59% as stated so I guess 10 hours of mixed use is about accurate.