Running out of pins isn't necessarily a reason to buy a mega. There are ways to expand pin count. Running out of code space or ram would be the usual driver towards a mega (and may well be the case here). That said at least on ebay megas aren't that much more expensive than a Uno
Depemds. If you are making only one of these it probably isn't worth it. You can use perfboard (which is what I usually do as I'm usually only making one for me) at much less hassle and expense than cutting a pcb. If you want to make hundreds then a pcb makes sense for only one perfboard is probably a better bet. Breadboard view supports various forms of perfboard so you can document what you have done
First, I wouldn't cut the wires off. They look to be standard 1/4 phone jacks just get an appropriate socket and wire to it leaving the petal intact so it can be reused if this doesn't work out. You do however need to figure out what the various wires do to know how to connect them. I assume this is intended to replace the pushbutton switches on the breadboard so two of the wires will be the switch contacts but you will need to figure out how that relates to the wires on the plug somehow and what the polarity switch they refer to does (none of which I know, nor does it appear on the amazon web page that I can see).
Still won't be driven from the Arduino, it is hdmi or VGA neither of which the Arduino supports. You still need to figure out where the screen is driven from (as I said I suspect that is from the Mac not the Arduino) and how to arrange that. I don't know either of those.