Does the engine have to be removed to replace the transmission?

Nope, but you will need an engine sling brace. here is a video of the sling begin used on a similar car, but the procedure is the same except you will need to undo the converter bolts and cooling lines. Also, I have included instructions for your car in the diagrams below.

https://youtu.be/6N6b5F2ChyE

Check out the images (below). Please let us know if you need anything else to get the problem fixed.

Not on that vehicle. You can remove it through the bottom while supporting the engine. However, it is a tight fit. You will need a support beam or cherry picker for the engine. Be careful when removing the axles, it's very easy to damage the seals in the transmission housing.