We want to use AES as a pseudorandom function. Given a random master key, we want to derive a set of pseudorandom values/keys, e.g say $n$ keys. Then from each of those derived keys, we want to derive a set of pseudorandom values, say $d$.
I think, one way of doing that is to encrypt $1,...,n$ using the master key. This gives us $n$ ciphertext. Then, we use each ciphertext as a key to encrypt $1,...,d$.
Question 1: In the above case, do we need to generate a random IV each time we encrypt a message. Does that mean we have to store all IV?
Question 2: If the answer to the above question is yes, then which PRF (that requires only one seed and IV) is suitable for the above case? Can we use the hash of each input, i.e. index, as IV?
Note that in the above case, IV does not need to be public.