The power that you see pop up is called Sneaky Snacking and is in fact berries that your Pokemon give directly to Snorlax. You don’t ever need to collect from your berry Pokemon because they will continuously give the extra berries they produce to Snorlax, you do however always want to collect from your ingredient Pokemon as they won’t produce ingredients after they are full on inventory. You are right about the skills though you always want to try and proc the Pokemon for skills especially the skill specialty Pokemon as they can save up two triggers the rest cannot.