If you use the stock browser that comes with it and have flash installed, you can watch Amazon Prime videos beautifully! I know because I have Prime and use it all the time. I think they must have updated the player or something because a while back I tried to watch Prime on my Galaxy 10.1 and it was horrible. Maybe the Note 10.1 has a better browser for Amazon Prime, but it works great.
You can download books and audible books using the kindle and audible apps, but there is currently no official way to watch the Amazon instant video on any android device. I'm really hoping this changes soon! I say there's no official way because there's no app for instant video on Android and it's not listed on the supported products by Amazon. That being said, I've been researching into this and read that if you go look for a download of Adobe flash (no longer available from Google Play, you have to download an old version and install yourself) and try to play from a web browser in desktop view with Flash installed, then you can get it to play instant videos in your browser while connected to the internet. You cannot download the videos and watch later (there's no Unbox app, which is required to download and watch later). Google Play does sell movies as an alternative.
Update--I can confirm that I was able to download the flash player apk file, install it on my device, launch the web browser in desktop view, and watch Amazon instant video in the browser. The quality was pretty good.
Yes, see here: http://forum.xda-developers.com/showthread.php?t=1774336 . Works with Firefox Beta and Flash 220.127.116.11, haven't tried Opera. Chrome doesn't support Flash. Install the Auto Full Screen Firefox add-on and you can watch videos in full screen without the nav bar.