Quick way to move items from inventory to chest
Solution 1:
You're able to take all items from a chest to your inventory by pressing X, but there's no way to do the opposite. You have to do it one by one. Mods won't work because you're on Xbox. I don't think there's any way to move items quickly from inventory to chest.
Solution 2:
No, there is no way to move more than one item from yourself to a container or follower all at once.