CSPRNG made out of Block Cipher running in CTR Mode? From Definition of CSPRNG, it has two characteristics


It satisfies the next-bit test.
It withstands 'state compromise extensions' - part of all of the state being compromised does not allow for reconstruction of the prior stream of random numbers.


The Design of CSPRNG can be based on


Cryptographic Primitives
Number Theory Problems
Sepcial Designs like Yarrow, Fortuna etc


If CSPRNG is constructed out of AES/Twofish/Camellia in CTR Mode, does it satisfy 2nd characteristic? i.e Compromise of internal state will not lead to reconstruction of prior stream of random numbers?

If we consider state as Counter only, or Counter and IV only (excluding the key). Will it be a CSPRNG? 

Will any CPA secure Block Cipher working in Counter mode made CSPRNG under the assumption that compromise of internal state only consists of compromise of Counter only or Counter and IV only (not the key).
If CSPRNG is constructed out of AES in CTR Mode, does it satisfy 2nd characteristic? i.e Compromise of internal state will not lead to reconstruction of prior stream of random numbers?


No, assuming you mean the CSPRNG is just AES in CTR, the state is simply the key and the IV, therefore compromise will allow trivial generation of both past and future outputs.

Exposure of just the key is also as bad, as you can decrypt the output using the key (in ECB mode) to recover the IV.