Transmission swap

Welcome back to 2CarPros. The transmission itself should be a bolt in replacement. As to which parts you will reuse, it will depend on what the seller includes. Many are set up as a drop in but it's possible that a sensor plug could be different. One thing to be sure to do is to thoroughly flush out the old transmission fluid from the cooler. You don't want dirty fluid running into a new transmission.

Is there a proper procedure for flushing the transmission cooling system before installing a new transmission? The old transmission is still in the car. Can the transmission cooling system be flushed thoroughly without flushing the entire system?

The normal way for someone without a transmission machine is to use compressed air at around 30 psi or so and apply it to one of the cooler lines. You blow out the old fluid then use some new clean fluid with some trans flush mixed in and blow that through in both directions, then use the air to blow it clear again. You just want to get any old fluid and grit out of the lines.