I'm back from the Apple Store and have installed Note Taker HD on a new iPad 3rd generation. It seems to work fine. As reported by a user this morning, the ink is just the same 1024x768 anti-aliased pixels displayed in 2048x1536, so it looks just as jagged (but a sharper version of the same...). The same applies to text, shapes, and images on the page.
The icons used in the app are all the iPad 1&2 compatible "@1x" style, not the high-resolution "@2x" style. The text on buttons and elsewhere in the UI and Help is sharper.
I only heard from one other user since the first one about compatibility issues (also about the ink), and it's been 18 or more hours since the first country with lots of copies of the app got their new iPads. That doesn't mean that there aren't other issues (there always are), but at least it hasn't been a disaster requiring an immediate upgrade...yet.
I'm already working on the next release and will give it better hi-resolution support. This will give you much nicer ink than before, as well as much better PDF viewing, text, etc. There are performance challenges with doing this, so it may take a little while to complete and test; however, the resolution can make an enormous difference if you are looking zoomed out at a page with lots of small written or typed text. People who upgraded to the new iPad will hopefully be pleased with the result if they were looking to take advantage of the sharper screen.
For those switching from an older iPad: If you are using iCloud and doing a backup/restore to clone the old onto the new, it may not move your files (at least I heard a report of that from one of my users). In that case, or if you aren't using iCloud or cloning the older one, use the Backup All tool in Note Taker HD on the old iPad to create an "archive.xxxx.nthd" file. Then use iTunes File Sharing or email or Dropbox, etc., to transfer that file to the new iPad's copy of Note Taker HD. Do a Restore All there. See the Help about Backup and Restore for more information. Update: I put detailed instructions on this page.
About the new iPad: Yes. Once you've read some text at high resolution, the old iPad looks like your reading glasses are a bit off, especially side-by-side.
I haven't opened the Apple TV yet to see if there are any AirPlay issues.