The hats that Jin wears are seldom secured by any kind of strap, even though they would need to be.
Since this game takes place during the first Mongol invasion of Japan in 1274, weapons (katana, wakizashi, hwacha) and techniques weren't present in real life until at least a century later.
Yuna says that the horses were scared off by the mongols, while Jin says that the horses will return. There is but one horse.