Who selects the organization that receives the adoption fee, the owner, the adopter, or Adopt-a-Pet.com?

That depends!

1. If a pet owner starts at Rehome.Adoptapet.com and creates a pet profile, that owner can select a shelter or rescue in Step 4's "how did you hear about us" question. That shelter or rescue will then receive the adoption fee if the adoption is completed and paid through Rehome. 
2. If the pet was not referred and the owner does not select any organization, Adopt-a-Pet.com collects the adoption fee and uses it to help all shelters and rescues.
3. If a pet owner uses a referral link provided by a shelter or rescue, the "how did you hear about us" question doesn't appear. The referring shelter or rescue receives the adoption fee.

(Rehome mails checks to referring organizations quarterly.)