I have read that you should try to run the hardwood so that it goes perpendicular to the floor joists to minimize movement, but also it is recommended that the wood be run in the longest direction of the room for best aesthetic appearance. For a 3/8" Engineered installation over 1/2 plywood over 3/4" plywood does it matter? Does one have priority over the other?

So your subfloor is 1/2" plywood with 3/4" plywood on top of that, correct? And the joist spacing is 16" OC, yes? If this is so, in this case, you can run your 3/8" engineered in any direction you wish. Just make sure to flaten your subfloor/underlayment.