I'm looking for a type (model) of NFC tag/card that would be capable of communicating with Android device and do the following:
Is there any NXP or other chip capable of doing the task?
The NXP chips J3A041, J3A081, J3D081 etc. will probably fit your requirements. As well as chips from other vendors like AT90SC20818RCFV, SLE77CLFX2400PM, S3CT9KA, SA23YR18, M7794 etc.
Will you develop your own OS for this? Looks more practical to use a solutions like JCOP, CONVEGO Join, JK21 etc. to use Java Card for card side development.
If you would like to concentrate efforts just on Android side - a ready tag like NFCCipher may work for you.