I believe ASE is using CPython running as native code, with some kind of bridge into the Dalvik VM.
I'm not sure what the status if JPython on Android is, but my understanding is that for many JVM languages it's non-trivial to port to Android, since many of them are expecting to be able to JIT to JVM bytecodes. This is problematic since Android doesn't use JVM bytecodes, it uses Dalvik bytecodes.
Don't know about it, but not sure why there would be lack of Python support. I suppose for standard apps they would have to run on a Java VM, so only Jython would be possible. Don't know about the state of Jython...
I assume if you root the phone, you can also run native apps, so why shouldn't Python work?
It works fine. Python (outside of the admittedly limited scripting SDK) is not a first-class API for Android. But it was running on the G1 within weeks of release (as was most of debian, FWIW) and there are certainly no barriers to using it.
But the supported API is Java-based, and that legitimately annoys some people even if it pleases others. Google isn't banning python, certainly, but neither are they pushing it. No different than any other phone, I guess.