How do I get the cheetah and/or white horse?

[Jaminos]

Does the quest unlock Coeurl for use - or do you need the seals (and chocobo) before you can use it as well?

The license allows you to use your mount. Regardless of the creature. All you are doing is paying for the ability to use a mount. Regardless if it is a Coeurl or a Chocobo.