which one will work the best?
I'd say they all have vices and virtues.
One improvement I have seen with the braid is the insertion of a bent SST rod. the Rod is formed into a shape that allows more coverage of the bottom of the tun and holds the braid in place.
I sue the braid and I manually cycle wort from the Tun to a pot and back to the Tun again to manage temperature and the rests. Then I batch sparge.
I have never had it get clogged.
and what size cooler is best for doing 5-6 gallon batches, and be able to fit enough grain for barleywine sized batches?
HUGE, as big as you can get.
It helps to pre-heat the Tun/cooler by filling it with hot water.
I started with a 5 gallon cooler and will be using a 15 gallon one soon maybe bigger.
You can also use more than one if you prefer more physically manageable units.