Why in cryptographic schemes we always assume that the key and plaintext are independent? Why do we always assume in cryptographic schemes that the key and plaintext are independent? What if the plaintext depends on the key? How can this endanger security? Is this assumption essential for both symmetric and asymmetric algorithms?
What if the plaintext depends on the key? How it can endanger security? 


Well, one trivial example where it can is if you are using RSA (without padding), and encrypt one of the prime factors. Someone seeing the ciphertext could immediately factor the modulus (by computing the GCD of the modulus and the ciphertext), thus endangering security.

There is a formal name for systems where this provably can't happen; circular security.  See this paper for a public key cryptosystem that can securely encrypt any polynomial function of the private key (up to a degree bound).

My personal opinion: If you really need to do this sort of thing, I think a hybrid scheme such as the Integrated Encryption System would suit your requirement quite nicely.