In order:



It doesn't really matter, because they don't come from the opposing trainer. You can get them even when you're fighting wild Pokémon.



A move being found mostly found on Pokémon of a different type than the one it has doesn't change what that type means (and Extrasensory is an actual Psychic-Type attack with the same property to boot.)



Agility's description is "The user relaxes and lightens its body to move faster," or, in other words, it does something mental and tells physics to GTFO (especially since it doesn't actually reduce the user's weight like the otherwise mechanically identical Autotomize does.) Sounds like a Psychic-Type move to me.