Thank you so much for helping out!
Interestingly you seem to have gotten things in a different order than I thought was possible, so maybe I have flawed logic somewhere else in the puzzle that enabled you to get Weavile and Houndoom, Sunflora and Tangrowth (since those two sets are pairs), and Venusaur earlier than I thought possible. Can you explain how you got those without the clues that come from the spaces at the bottom you currently have unsolved?
The next step necessary is indeed requiring some level of Pokémon type effectiveness. Qwilfish (Water/Poison) resists both of Volcarona's types (Bug/Fire). Therefore, Volcarona is in an Ultra Ball, per Qwilfish's rule. Every statement you wrote is also true.
Also, as an aside, should I remove the typings listed? Since type-effectiveness is already required "specialized" knowledge, though very easily found out by consulting a single type chart image, this puzzle does require more than pure logic. So it's more trivia-ish I guess to some degree. Taking out the typings will make this trickier for the actual Pokémon fans (though probably will result in many people not being able to play.) Thoughts?
Again, thanks for the help.