Bizzaro Playing Cards
Note that (1) and (2) are satisfied by any derangement of the cards. There are probably many "mappings" that satisfy your conditions, but since all you seek is one possible answer, I propose the following solution.

For the values, map them in this way:

A -> 2, 2 -> 4, 3-> 6, 4 -> 8, 5 -> 10, 6 -> Q, 7 -> A, 8 -> 3, 9 -> 5, 10 -> 7, J -> 9, Q -> J, K -> K

Then: A234567890JQK -> 24680QA3579JK

So clearly any straight will not map to any straight. And because any 4 of a kind maps to a 4 of a kind, there won't be any suit duplication (your condition 3).

For the suits, consider some order of the suits [clubs, diamonds, hearts, spades]:


For values A-4: map to the same suit
For values 5-8: map to the next suit (e.g. clubs -> diamonds, spades -> clubs)
For values 9-Q: map to the previous suit
For K: map to the "opposite suit"


Then any flush (which consists of 5 cards of the same suit) cannot map to a flush, because you cannot obtain more than 4 cards of the same suit if you start with a flush.

Note that changing the suit of the K resolves the issue where K -> K, since it is required that no card be its own "twin".