My wife was trying to unplug the sink with a snake-like brush and the old trap crumbed. As I tried to remove the rusty old trap, the connection to the sink stopper broke and so I was forced to yank the stopper as well. It was so old that I couldn’t get it to unscrew. This video helped give me some insight into how to remove the sink stopper. I got the nut under the sink to budge and create a crack between the gasket and the nut. However, I could not get the drain to unscrew and I resorted to use a hack saw to allow me to pull the sink stopper.
Thanks to this Old House for posting the video to help me solve my problem. The final product shown on the right above.