It can depend on what you're showing your horses in. Generally speaking horses with high levels and higher PS will be more valuable. Training Boost can also play a part in their value. If you're showing your horses mainly in discipline shows these are pretty much the only factors you look at. 

1. how i see which horse are best for Hunter in my stables? 

A. There really isn't a good way to see exactly what horses are best for hunter without individually looking at their stats. The stats for Hunter is Intelligence, Agility and Flexibility. So you'll want to look at each horse to determine if they're high in those three stats. However, if you're dealing with foundation horses or horse with very low PS, you could just set them to Hunter without really paying attention to that. It doesn't matter as much with low PS horses because within a few trainings they'll be geared towards whatever you'd like. I only consider what is best for each horse when they have PS over 500 to 1k

2. But training these horses does affect very hard their stats? Or these stay the same as day born?
a. Correct! Training affects their stats like I said above. You train your horses in whatever discipline you would like and that increased their PS. Training Boost is also important because it acts like a multipler when you're training your horses. So the higher the Training Boost, the more PS they'll gain in their lifespan. Training Boost comes from the parents Levels. 

Hope that helps!

Thank you Roze!

Very detailed info, very thank you!