sorry for not rplying to your stock fitting question. I will try to find an answer for that on Wednesday.
A friend had his own clay thrower, and he shoots it on his own land. I am not sure if you could shoot them on BLM land. I'm pretty sure you'd be okay in the desert, or deep in the woods.
Regarding unsupervised practise, I doubt that any clubs would allow that. At RR&G club, it can often be quiet on a Wednesday, and there is a chance that there may only be one or two shooters on the field.