I've had no problems with aftermarket radiators on my LSes (strangely enough).
It's just a radiator. The transmission is cooled by an oil-to-air combination cooler between the radiator and the AC condenser. The power steering cooler is a separate section of this combination cooler. There is also an oil-to-water cooler, but it is external to the radiator. It's at the oil filter adapter.
The recommended procedure is to remove the radiator from the bottom of the car. I found that to not be too difficult, and it looks like doing it from above would be much harder. With the front wheels up on ramps, there was plenty of room to do it from the bottom. Just tilt the radiator slightly after it is most of the way down.
2006 Lincoln LS Workshop Manual
Note that the whole manual is here:
2006 Lincoln LS Workshop Manual -- brought to you by Owlman