Generic Cookie Names

A custom cookie persistence profile with a specific name will not include the pool name in the cookie name - only the encoded IP and port. The default cookie persistence profile has the added benefit of being usable across VIPs (with the same pool), so your custom cookie profile will simply use the pool that is configured in the VIP. Otherwise your custom cookie profile should be fine.

As for encrypting the cookie contents, you could simply enable cookie encryption in the HTTP profile and provide the name of the cookie(s) that you want to encrypt and a passphrase.