Vet. He could have an abses or something
If the vet and the farrier clear him he might be traumatized from something. Try doing some horsemanship and attemtping after sessions.
Yes farrier told me to tap on his hoof or just squeeze the chestnut like you wanna pull it of the horse will for sure lift his hoof. You dont need to use extra Money than whats needed so wait longer and talk to the farrier if needed
Taking him to the vet would be a good idea considering that there should be no reason for him not to pick up his hoof unless it is sore. That being said, my mare used to be a monkey for picking up the exact same hoof and still gets a little fussy, but I just continue to ask her to pick it up and leave her alone when she does.
If there are no underlying problems, such as soreness, a little bit of re-training through positive re-enforcement may help!
I've been keeping an eye on his frog. And it seems to be okay. I might to take him to the vet. He doesn't limp, and doesn't act like he's in pain.