I have very tight shoulders. I take a very wide grip on the bar while squatting (palms of my hands out on the collars). What would the best stretches be to improve range of motion? What is the best rep/set and frequency scheme? How long should each stretch be held? Is it better to hold a stretch or is a dynamic approach better - or, is it combination of the two? Are there specific medical protocols for increasing range of motion?
You can try doing something like this: https://www.youtube.com/watch?v=yV-jQmVUEWk
Otherwise, there is no specific or “best” stretching protocol for this. The more you try to get the bar down into the right position with a narrower grip, the more this should improve over time. With that said, not everyone is able to achieve meaningful improvement, and some folks just end up needing to high bar squat.
As a follow up, when I do what the YouTube video shows, from a pain or discomfort perspective, is there a point at which I should not force things? When I do the stretch as shown on the video hard enough, it becomes uncomfortable. Would it be accurate to say as long as I’m not feeling a sharp pain I could go as deep into the pain cave as I can tolerate (without doing something bad to my joints)?
Yes, it is going to be a bit uncomfortable. That is OK. I would not think of it as expecting to go “deep into the pain cave”.
Most importantly: You are not doing any damage to your joints.
What are we trying to stretch with the Horn stretch? Muscles or tendons or both? Which ones?
I unfortunately do not have time to lay out the detailed anatomy of the shoulder in the context of this stretch, sorry.