What exactly does the user have to do for the referral to count?

It says this, These are the users who've listed you as a referral but haven't quite finished the process. To count they have to adopt a pet and have at least 10,000sP on their account!

By adopting a pet, does that mean adopt one from the pound or make one?

How long does it take for the referral to count?

Last I heard, referrals go through every night at midnight, so they should go through the same day. As for what they need to do, we don't know exactly. Staff hasn't said.

Make a pet and have 10k sp. I usually make sure my friends have these requirements by making them go to both the pet page and the User Dashboard. Referrals go through at midnight, so you have to wait until the next day after they've done the requirements.