Pesachim 60: Can You Bring a Korban Pesach on the Wrong Day?!

Improper intention invalidates the Korban Pesach. Note 4 essential component parts of the sacrifice. The degree of intent here is specific - and yet the case this daf considers is one where correct intent is there together with mistaken intent. Namely, that one could intend for the sacrifice before him to be the Korban Pesach, and also have intent for it to be a korban shelamim... How do those different intents affect the korban? What about a Korban Pesach that is slaughtered with correct intent, but on the wrong day? This seems an extreme, boundary-testing case. Plus: How does one show intent?