If an ability triggers whenever you activate an exhaust ability, that ability resolves before the exhaust ability resolves.

If an exhaust ability of a permanent is activated, and then that permanent leaves the battlefield and returns to the battlefield, it becomes a new object so its exhaust ability can be activated again.

Multiple instances of ward each trigger separately.

Exhaust abilities can be activated any time you could normally activate an ability.

If an effect causes Winter to become an artifact, its second ability will cause it to gain a second instance of ward.