Theorem 3.1 in Peter May's "A General Algebraic Approach to the Steenrod Operations" gives a better behaved definition for $Sq^k(x)$ for (co)chains $x$ which are not assumed to be cocycles. It probably makes sense to think about these questions (Adem relations or Cartan formula on the cocycle level) using them rather than just the cup-(n-k) formula. I find it highly implausible that either of these (Adem relations or Cartan formula) should hold 'on the nose', no matter what definition extending the usual one on (co)cycles one gives.

Theorem 3.1 in Peter May's "A General Algebraic Approach to Steenrod Operations" gives a better behaved definition for $Sq^k(x)$ for (co)chains $x$ which are not assumed to be cocycles. It probably makes sense to think about these questions (Adem relations or Cartan formula on the cocycle level) using them rather than just the cup-(n-k) formula. I find it highly implausible that either of these (Adem relations or Cartan formula) should hold 'on the nose', no matter what definition extending the usual one on (co)cycles one gives.