I believe it is considered a significant break in a block cipher if you can distinguish messages encrypted via that cipher against random bits constructed in a similar way (i.e. constructing an adversary such that it has a non-negligible advantage in the PRF security game against the family of functions for the block cipher).
You can tell mode somewhat reliably (I've hit 75% confidence or so) by the entropy content of the ciphertext, and I've never done a Markov chain longer than a native word so I'd assume that gets better the longer your chain is. I would also assume people much smarter than me have compiled heuristics that are more useful than just determining the mode.