The connection to a laptop screen is usually LVDS, which is not a common connector in PC-land, but is a standard signalling method. You'll also need to power the backlight, which is pretty straightforward. You'll need longer cables for both of those.
You'll probably also need to fabricate a bracket to actually mount the screen. I can't recall any laptop I've disassembled having a mounted screen; they were all held in place by the monitor half of the case, not screwed to the case, but held in place when the case itself was screwed together. It was a pressure fit, inside. I guess it would depend on the make & model; some probably do screw it down.
If you get a case with a deep enough lid, you won't have to worry about it. I don't know of any joysticks that aren't meant for, or designed to look and act like they were, designed for use in public places. Accordingly, none I've ever seen have easily removable sticks. An easier method than retracting would be rotating down and away from or toward the latches. Those mechanisms are tricky. It would be even easier if you had a mounted panel, but, when opened, placed controls INTO the panels and optionally fastened them down. Kind of like how X-Arcade joysticks (ugh) sit inside X-Arcade cabinets (ugh). Removable, but sturdy when in put in place.