Does routed heroes are allowed to train or to rest?

I think you are referring to commanders that become routed in spring? Since as far as I know that's the only reason a hero would be routed during the summer (when training/healing occurs). Other causes for routing would be quests, and would have taken place either last year (and you'd have become unrouted in spring) or this year, but then you would have quested and not been able to train/rest.

I THINK that routing only prevents your unit from moving, not from training or resting.

A Hero can also become routed with the Revised rules when it supports in battle and his army loses. So this can happen with any hero in the Spring if they support a battle in the Spring.

If a hero is routed in the Summer, it can still train or heal. Only the "Move" option (which also allows you to do a quest) is prevented, as described on page 26.