Does the wire purpose have to travel with the actual wire request? It came to my attention that the wire purpose is being input in the reference field of the wire when processing. We had an international wire recently to a country that is known for being hostile to religious organizations and this wire was for financial support of such an organization.

You only transmit the information that the customer requests you to transmit. If you choose to gather wire purpose for your own internal information, that has no place on a wire transfer order.

Also, what criteria do you apply when requesting documentation to verify the wire purpose?

Some countries; e.g. India, require a statement of purpose on incoming wires. Some correspondent banks also require a statement of purpose on the wires they agree to transmit.

What you require as "proof" is purely a matter of bank policy, but the general answer is an invoice; i.e. something indicating the originator owes the beneficiary the amount of money involved.