Steve is right...these oil coolers do have play in them on the rubber grommets, totally normal. Just make sure you're not missing any of the metal inserts that sit inside the rubber grommet or you might have too much play.
Draining the transmission fluid, I usually insert a folded shaped piece of thick paper or cardboard to funnel it off the exhaust pipe otherwise, it does what happened to you. At least that way you can direct it towards an oil pan.
