Nulon synthetic trans fluid is fine, just don't use dextron III, it will slip and ruin your clutch plates. Back flushing is a good idea, as is dropping the pan and cleaning the filter if you are keen.
Mixing antifreeze is a bad idea, but who knows what is in your cooling system now... I would flush it a few times with water and drain it completely, (the block drain plug is a pain to get to) then refill with a decent off the shelf coolant.