I used Motorcraft dual clutch transmission fluid when I changed mine. It took 3 qts. For the diff I used Valvoline in the squeeze pouch that you can get at Walmart. It was 2 qts. I think but just fill it until it start to run over.
Regardless of oil, the MT-82 in my 2015 is much better than the ones in my 2011 and 2013.
And I had a GT350 and I didn't really see the big deal about the Tremec. It used to pop out of first gear from a stop here and there- quality! The Getrags never did that to me, regardless of the abuse I threw at them. Only thing that ever broke was the weak stock clutch disc when I raised rev limit to 7800. It just couldn't handle the revolutions and locked up on the throwout bearing when I shifted at 7800 rpms. I learned the hard way that these cars are not just made up of components, but are SYSTEMS. Turn up the juice from stock and you'll find out what shakes loose the hard way: the weakest link in the system breaks.
In summary: I've never had an issue with stock fluid and see no reason to swap it out. If the transmission is maintained, any shift issues people notice are likely due to a weakening clutch disc and pressue plate- makes your synchros work harder when the clutch is slipping more/wearing out. I suppose you can double clutch like Vin Diesel (so Fast, so Furious!) or a semi driver. But the proper solution is a beefier clutch.
I've yet to mess with the rear end on this car- and likely won't- but I used to run STP NON synthetic 75/90w in my S197s. It really depends what kind of carrier you have; you'll need to investigate for your car:
If you have a Ford "traction lok" carrier, make sure you add friction modifier in with the gear oil (or buy something with friction modifier in it) so the carbon discs in the carrier don't lock up.
Other carriers do not require slippage/friction modifier.
Example: I used Eaton True-Trac carriers (aftermarket) for better power distribution when cornering. That's a helical gear system (not locking carbon disc) that pits the passenger and driver's side axles against each other to distribute power from the drive shaft; using synthetic and friction modifier is the worst thing you can do on a helical gear type of of system- it will make the carrier gears slip and chew them up.
I would confirm your car's specs/setup and proceed appropriately.