How do I test my horses' bravery? What is this rating good for? :)

You should go to Services > Explore. Go until your horse spooks and it will tell you. It isn't really good for much in showing to my knowledge, but people breed for WC Bravery :)

What Primal said about exploring to discover the bravery!

Once your horse has been tested for their bravery rating, you can buy a roundpen and train it, and then explore to find decor and maps that can be used when capturing wild horses. Or you can just buy a fully trained bravery horse and explore with it- roundpens can be expensive if you're not actively breeding and training for bravery :)

Another question, so the horse is born with a bravery level and training doesn't change that? Training just helps them to Spook less often? So test them and then train the ones with higher ratings for exploring?

Yes, Mountain