You can have your coolers hot flushed, I would not do that to the trans side. This is typically done by hooking up to the lines where they come out of the bellhousing and hot fluid is sent through the coolers in both directions in different patterns to dislodge any debris.
This is 100% fine.
As for fluid it honestly depends on year. The GM spec fluid for all years is Dex 6 but initially the 5spds all received Dex 3. All of our built transmissions are recommended with Dex 3 equivalent fluid. There is nothing wrong with high dollar synthetic fluids, they will lighten the wallet more tho.