The fluid is drawn from the pan, through the internals of the transmission, through the external cooler and then back into the pan.
You want to do the pan first so that all fluid drawn up is fresh and because you disconnect the cooler line you stop any of the old fluid that was in the transmission from going back in and mixing with the new.
So yes after you do the drain/refill you disconnect the cooler, start the engine and let a litre out, stop the engine, put a fresh litre of fluid in through the dipstick tube and repeat untill the fluid coming out of the disconnected pipe is nice and clean.
Then check the fluid level is within the cold markings as a guide and then drive for at least 15 minutes and then check the level is within the hot marking. You shouldn't rely just on the cold markings as they are just a guide, the hot marking are the accurate ones.
To check the fluid level make sure the engine is running and move the gear selector through each position (pausing a couple of seconds between shifts) until back in Park and then check the dipstick (keeping the engine running).
Sorry I'm not sure about the gasket/sealant for the pan, may be worth searching through the forums on the US LOC site - many more LS, GS and SC owners in the US