Will Elites and Premiums still max out with the special clinics? Also, will a horse under a level 1 rider always be level 1 or does the rider have to level up for the horse to level too? (I hope I worded that right, sorry if that second question's confusing)

I'm not sure about your first question but re: the second one - horses and riders train and level up independently. Horses with an e or p rating are likely to level up before their rider so will often need to be reassigned to a new one when they level up. :)