The Greeks weren’t really noted for their horsemanship and a man can outrun a horse anyway.

However having fresh horses at properly spaced outposts like the Romans used are far superior.

Yes, but at this time only your very wealthiest members of the aristocracy would be able to afford one. That part of greece is too mountainous to easily breed and take care of horses, so they became more of a status symbol than anything. It's one of the reasons why Macedonia was able to conquer the southern Greek states with relative ease.

Horses actually just walk the whole way. You can do about 30 miles a day.