That's correct, it's an auxiliary cooler. Looks like a small radiator in front of the engine radiator. Lines from your trans carry oil to the radiator, then the cooler and finally back to the transmission. The tranny cooler on your E Series is more than adequate for the job. There's no need to upgrade. Now, as far as intake. Yes, I'd upgrade to a fresh air intake system like a K&N filter charger. The more the engine breaths, the better. You'll really notice a difference in the winter. Cold air injected into the throttle body creates an increase in horsepower.
2013 Jayco Eagle 328 RLTS
2013 Ford F350 6.7L 4x4 CCLB W/Air Lift air bags
2002 Coleman Bayside Elite (traded)
Me, Momma and Charles (RIP buddy)
And introducing Sofie!! ;)