How to prove that a ciphertext is encrypting multiplication of two values?
Yes, this is doable.  You can verify that $m=a \times b$, given El Gamal encryptions of $m,a,b$, since El Gamal is multiplicatively homomorphic.  You can also prove that the El Gamal decryption of some ciphertext is in a given range (using a combination of Schnorr proofs).  Putting those two together will give your first desired primitive.

This requires you to use El Gamal encryption rather than a stream cipher.  However, if this is a problem, it is also possible to do it for other encryption schemes as well.  It is known that every statement in NP can be proven in zero knowledge using generic (but slow) schemes, so this can also be extended to work for any stream cipher.  The advantage of ZK proofs based upon El Gamal or other public-key schemes is that they can be made more efficient.

See, e.g.,


Proof of correctness of a homomorphic ElGamal sum
Is there a public key semantically secure cryptosystem for which one can prove in zero knowledge the equivalence of two plaintexts?
http://www.win.tue.nl/sedan/sedan2009/SchoenmakersSedan2009.pdf


There is tons written on this subject in the research literature.  You'll need to do more research on your own to learn this subject.  You should spend some days studying basic material on learning elementary zero knowledge proofs, first by reading textbooks, then by reading some of the classic papers.  A good starting point would be to look at textbooks on theoretical cryptography, such as
Rafael Pass and abhi shelat's lecture notes and Wenbo Mao's textbook.  You could also read http://www.austinmohr.com/work/files/zkp.pdf