Depends on what's leaking on the turbo. If it's coming from one of the fittings or flanges, you can fix it. If its coming from between the bearing housing and the compressor housing then it's time for a rebuild.
The computer will have identification labels if it is a toyota one. Is it idling with the O2 sensor hooked up now? Check codes again, it may have a different code now.

No codes yet. It was driving fine for a min but the toggle switch for the power went. I'm about to head to advanced to to grab one. Wish me luck.

In your last photos:

1) Fuel resistor pack relay connector. This is normally run to a relay which switches the fuel pump power to a resistor pack to help lower fuel pressure at low loads. It's commonly deleted with aftermarket pumps although this requisite has been largely disproved.

2) and 4) A pre89 ECU hacked into an '89+ chassis?

3) The HAC sensor connector. The HAC sensor is used by the ECU to account for changes in altitude. If you live near sea-level and don't drive in different elevations you may bypass it with a couple resistors, otherwise pick one up (used or new). In '89+ ECUs it is internal. See the diagram linked to below for the HAC bypass resistor values (Just a box on the side):
http://www.fadingworld.com/CRE/Autom...087-88-r21.jpg

That is one terrifying wiring job and the mods appear to be half-assed at best... I wouldn't have bought it for anything more than parts.

Would you say it'd be best just to sell it? Or would you say its a pricey fix if I do the labor?

That's up to you. If you like the car, then keep it and fix it. If you're not too into the supra, sell it before anything else breaks. Remember, it's a "highly sophisticated" 20+ year old car and yours has been obviously butchered by someone. It will take time, research and money to build a nice mk3 supra, probably more than you'll wish to spend. It will be pricey no matter who does the work, in the end you get more for your money doing the work yourself. If the body and interior are nice, I would fix it up for sure.