I don't like talking about "the best". It depends to the needs and other things.
Maybe other physic wrappers for Mogre are available, but also it's a question of documentation.
Without tutorials, examples, member descriptions etc. it's difficult to use it.
Newton 2 is in development for many months now. If I remember right, the main work is done (API revision, optimization, etc.). Now there are only tests and bugfixes.
Just try the open beta version and look if it's fine for you.
If you find bugs, post it in the open beta
I don't know what you mean with false positives/negatives
related to Newton.
Maybe it's a question of configuration?
You want to have better acceleration functions. I don't know about, but maybe other physics engines doesn't offer "better" ones. It's possible that your wishes aren't available directly by other physics engines, too. It could be that some things have to be done by the application developer. But I don't want to make a discusstion about here.
A good place for general Newton questions (independent of Mogre) is the OgreNewt forum.
Also you can talk about this in the Newton forum itself.
There are many experts who can help you in detailed questions.
And if you miss something you can tell a whish. Maybe it will be included.
For my application I had a special need and the author of Newton did include it.
Newton 2.0 Archemedia Open Beta: (Newton Website)