why is double-spend not an issue for payment channels?
Great question. In order to understand how it works, you have to understand that opening a micro-payment channel (MPC) requires a commitment before you get started. That is, both parties have to show proof that they can't double-spend these transactions before they get started.

How is this possible? SCRIPT, of course. A transaction (2-of-2 output) that's dependent on other transactions is signed first, which opens up the MPC, and then, and only then, do both parties release the transactions that would make this other transaction valid. At that point, the funds are committed to the channel and cannot be double-spent since it's locked in a 2-of-2 output, which must be signed by both parties, hence eliminating the possibility of a double-spend.

A more thorough explanation can be found here.