no

Yes he is

He absolutely is not. He is a permanent pool unit.

so is he limited or not? ,

Rain is not time limited

Rain is standard. Fryevia is limited.

This is easily confirmed by going to World Quests > Glacial Lands > White Wolf's Shrine. You will find Rain's unit shard quest, there. These quests are only available for standard units.

Hope this clears up any confusion!