The following questions are of pure theoretical nature. I don't have an application in mind.


Is there a key that makes AES the identity function?
Is there a key that makes AES the identity function for certain
inputs?
Is it known whether such keys might exist?
Are such keys known for other ciphers?


For example, I want to encrypt "Hello StackExchg" with AES with said key and the ciphertext should be again "Hello StackExchg".

When I encrypt "Hello StackExchg" with the perfect cipher one-time-pad, there is a possibility that the resulting ciphertext is again "Hello StackExchg". This possibility must exist (though it is unrealistically small), otherwise, the ciphertext would not be perfectly random but depending on the plaintext. The corresponding key for one-time-pad would be all zeros.

In the "ideal cipher" model, the block cipher is a permutation of the space of input blocks, chosen uniformly among all such permutations. A plaintext that gets encrypted to itself is a fixed point for the permutation; it is expected that about 63.21% of all permutations have at least one fixed point (a permutation with no fixed point is called a derangement).
Thus, assuming that AES is an ideal cipher, then it is expected that for about two thirds of possible keys, there will be at least one plaintext block that is encrypted to itself.
Now, finding that fixed point is quite another thing; we in fact expect not to be able to easily find that fixed point or even ascertain whether it exists or not -- the "moral" reason is that finding a fixed point on a random permutation requires hitting it exactly, which is akin to a brute force on the whole input space, of size 2128 in the case of AES. In other words, if we could find fixed points or even prove (non-constructively) their existence or non-existence for any specific key, then this would mean that we could differentiate AES from the ideal cipher model, and that would be worrying.
(See also that answer on crypto.SE.)
Note: while a majority of AES keys should have at least one fixed point, it is strongly expected that there is no key that turns AES into an identity function (i.e. all plaintexts being fixed points): there are 2128! ≈ 22135 possible permutations, a number waaaay larger than the 2256+2192+2128 possible AES keys, thus the identity permutation is very unlikely to correspond to a key. But then again, we don't expect to be able to prove it.
