My warehouse is currently full with 30 different types of items. However, none of my items are even close to the 999 limit. When I go to Quests I see a warning that reads "You don't have free space in your Inventory. Your pet won't be able to take any rewards except money!" 

My question is, does this only apply to new (to my inventory) items? Or will my pet not be able to add, say, more vermillion to my existing vermillion slot?

You won't be able to get only new items in this case. I edited this notice a bit to make it more clear. Sorry, it was really confusing.

Thanks for clearing that up!