No you do not have to remove the urethane.

You will neeed to sand with a fine sanding sponge and wash with ammonia and water to make sure they are clean and the primer has something to bit to. Then you will need to prime them with an oil based primer and you don't need to use a fast dry type unless you need to finish paint them in a couple of hours. The finish paint should be an acrylic semi-gloss or gloss depending on your preference and plan on two coats of finish paint.

Thank you.

I also want to mount decorative moldings on the existing hollow-core interior door. Can I just attach the molding with glue, since I do not think I drive any nails?

after the prep, but prior to priming, use a good glue like gorilla glue or an epoxy type glue to fasten the moldings. it may be necesary to remov the doors so that it is easier to apply the moldings on a level horizontal surface rather that a vertical surface

JMHO

Carl