Could this stop double-spending? Let's say you have a 15 second block time, and after your first transaction, your wallet's send button is frozen, and you have to wait for 2.5 minutes before you can send another transaction.

Would this prevent somebody from doublespending?

BTW: in case you want to ask how this could be enforced on the send button: Hmmmm maybe with something like scrypt which is timebased.
Double-spending has nothing to do with double clicking the send button.

It involves a malicious user, who will not use the standard client. It will hack one to avoid any client-side limits, sending two transactions to a different side of a blockchain fork, so both sides don't know about the transaction sent to the counterpart and includes the transaction in their forked version.

If you manage to do so while using both transactions to pay for something and are accepted, after the reorganization (the process by which forks are resolved) only one side of the fork will be mantained, so the blockchain will be OK as only one transaction will be kept, but you got both payments effectively using the same coins to pay two different merchants and one of them will lose what you paid.

As you can see, it is not an usual scenario. Forks are more or less common but not really deep as they typically get resolved within a single block. That's why it is recommended to accept a transaction only after six confirmations, as a fork of six blocks is almost impossible to appear under normal circumstances. But do note that even this has happened, when a bug between software versions kept some miners from agreeing to mine on what would otherwise have been the consensus branch.