[Blue Dog]

You are correct in assuming the auxiliary cooler is installed after the OEM cooler.
As far as which direction to mount the inlet & outlet hoses, it really doesn't matter.
You want to make sure you don't have any kinks that will restrict flow in the hoses.
I mounted my cooler with the connections on the side, with the inlet at the bottom & the outlet at the top.
That way there is no possibility of air being trapped in the cooler, which is highly unlikely in any case because the oil is pumped through the cooler.
When you cut the oil transfer lines, don't be surprised if the oil has drained out of them.