Best oil for engine, gearbox and diff?

I have a T5 in my Eagle and run synchromesh lube in it - I tried other stuff, and even AMC changed specs at least 3 times and settled on a lube that synchromesh from Penzoil matches in specs. In other words, it meets the Chrysler specs which were based on their own and AMC specs - so it's pretty much a modern version of what AMC specified for the T5 and 4 speed from the later years. 
Early cast iron old-school transmissions should use the specified lube.
Use a good hypoid gear lube in the differential. Pick a top brand - although most of them meet the specs of these differentials today. 

LOL - that bit about synthetic is BS to put it nicely. I run synthetic in my Eagle....... 
There's no such thing as "made to run on........" as far as oils in these engines. Modern oils meet those specs and synthetics often beat the heck out of those old specs. Many of us, and I mean many, run synthetic oils in our engines from the 60s, 70s and 80s.
I have for years. 

As far as the viscosity or "weight" - 5w30 is best. That comes from an engineer, member of SAE, a guy who does testing of oils.

Here are some quotes from his blog a list of some of the best oils by his testing (WEAR PREVENTION which is all we care about - oil is to prevent wear so if an oil prevents wear to a higher degree than others, then it's a good oil - pure logic) - and the link to his blog.
The PSI ratings are the pressure an oil takes before it starts to allow contact and thus wear. Higher PSI is better, REGARDLESS of any other additives, etc.