What is this type of car called?

I would say why the hell not! All convertibles had torque boxes in the frame to make it more rigid. I would weld a temporary structure in the car while you remove the roof until the frame is beefed up. It will make sure everything is aligned correctly when you cut the roof off. If its a car that is in rough shape I think its a fun idea. I've done it before on a vw bus. It was a blast we put 8 bucket seats in it and cruised the town. It came out real nice but hey I'm in California!