Wow Christina you appear to have cracked it !
The first thing that strikes me with this series is a general improvement in image quality from previous attempts, no idea why but it's noticeable.
It looks to me as if the 1/60s for the speed the horses are travelling is just about spot on for what you are trying to achieve, I do not think there is anything to be gained by shooting slower, you will just get more failures. 1/80s is going to give you a slightly better keeper rate but at the cost of losing some of the motion.
With regard to having the jockeys or horses head in focus I suspect ideally it needs to be both and perfect examples are No's 1 & 4 but achieving it I suspect comes down very much to luck although they are likely the objects of least movement out of sync with the camera.
As for No 2, the focus is not just sharp on the jockey at the back but also on the right flank of the front horse No 3 suggesting that your DOF is spot on and anything 'fuzzy' in between is not actually out of focus but 'motion blur'.
I'm not sure if you know where your shooting position is going to be or you have any option in the matter but for my drag racing I measured my position in relation to the track/cars and worked out the DOFs I would get at varying apertures to give me some confidence of how tight focus would be. For measuring I used google earth, surprisingly accurate and a very worthwhile exercise.
Not sure about the tripod but it's certainly worth a try, but the fact that where these are sharp where theres no motion blur shows your panning is pretty good.
Well done
Grahame