The amount lost from the cooler will be minimal. If the trans had enough fluid before disconnecting, it should be OK to drive but check with a stick first.
Some get all in a fluster that the level must be within a teaspoon, then forget what happens when the car goes up a hill and fluid sloshes to the rear of the trans.
Mixing the same type of fluid isn't an issue in small amounts and likely not in larger amounts as all fluid of the same type is theoretically mixable.
One of your lines might not have washers. Have a look at the pic, this type does not use washers. I'd have to see a pic of your current setup to determine where any washers go.
The complete fitting guide is available at
https://www.discounthydraulichose.com/v/vspfiles/downloadables/thread_guide.pdf