Well, i didn't searched for sound, particle system and ai in particular by myself. So cannot say for sure.
About physics: i tried to used BulletSharp and it was successful (at least for small sample that i made), i do not added it to Lime examples because it requires bulletsharp.dll which is 1,5mb (not small just for one small example). Cannot say anything about performance (about comparision in c++ since never used it, can say only that its possible to use it and it works

) The code to quick look
http://pastebin.com/N75qzpzp
Lime does work with .NET 4.0.
There is no tutorials, only examples (on C#) provided with SDK.
For documentation you can use almost on 95%+ Irrlicht native documentation link -
http://irrlicht.sourceforge.net/docu/index.html , Lime also has IntelliSense xml file, which VisualStudio 2010 uses automatically and help you with types. Not all namespaces documented, no intellisense for Scene and GUI namespaces.
About stability. Well, i didn't use Lime in big projects yet, so cannot say for sure. I can say is that all examples works fine, they are stable and has no memory leaks. Anyway, you should trust anything i say about stability, if i would you and decide to develop something big on something new, i would rather test it by my self by writing some small project, check features and performance, if it satisfies me.