Welcome to the club (I also just bought a white 245, a '93 automatic).
I'm curious if this is an LH2.4 car or LH3.1. 3.1 was used on manual trans 240s starting in '90 I believe (I had a '90 244 5spd that was 3.1, and my '92 244 5spd was originally 3.1 but converted to 2.4 to add a turbo using the factory LH2.4 computers).
The ECU located behind the panel on the the passenger side kick panel to the right of your feet when seated, will be -572 if it's LH3.1, or (I believe) -561 if LH2.4. The white label ECUs are better than the brown or pink label, and the later -951 ECU was the better/best LH2.4 non-turbo ECU as I recall, but a '90 probably has one with a '5xx' number as the last three digits of the p/n on the label. You can also ID the LH3.1 system by the air mass meter that will have a 4-pin plug instead of a 6-pin plug (Only 5 pins actually used in the 2.4 harness). And the throttle position sensor is different, I believe the p/n ends -001 and it's a potentiometer style instead of a click on/off type. LH3.1 is fine when it works, but the brown label ECU might be a weak point so do look for a backup. I acquired my '90 sedan for $250 because it had a dead ECU and the owner couldn't afford a new one and couldn't find a used one (took me almost 6 months to run across one).
Head gasket -- it's probably fine. But do investigate the coolant discoloration just in case. Could just be rust, etc. discoloration if a previous owner ran straight water for a while and then switched back to using coolant.
The breather box on the intake side, under the intake manifold, can be a source of oil leakage. Could be disconnected or broken vac lines under there, too. I've also had a front cam seal pop out and dump oil down the front of the engine. The valve cover gasket may leak down the exhaust side, my '87 cylinder head was BLACK on the exhaust side from oil leakage + road grime collecting. Give it a good cleaning and then look for the leak(s).
There's also a big rubber/metal plug in the back of the cylinder head that can leak, but that just dumps down the back of the engine. That plug usually come with a valve cover gasket set for the '85+ engines. I will usually give it a light coating of grey RTV around the edge when installing it. If you do end up taking the head off, you can fabricate a metal strip to screw into the back of the head that holds that plug from potentially popping out (IPD sells the metal piece but you can easily make one for a lot less). The valve cover gasket goes over the front and rear cam caps and at the bottom of each cam cap the gasket makes a sharp bend...that's a spot where I will dab a tiny amount of grey RTV at those corners to help it seal. Some people are very against RTV, some over do it. I try to use it sparingly.
An alternative that I like is Hylomar blue, a non-setting gasket sealer (credit: MikeP!). This stuff is great on the thin paper water pump gaskets because you need to slide the pump up against the underside of the head to seal against a rubber o-ring gasket, and that sealer allows the pump to move and also ensures you won't have to take it off because the gasket didn't seal. Speaking of the water pump, the square-profile o-ring that goes on the metal water pipe and fits into the back of the water pump is another place I've used a light coat of grey RTV.
The M47 trans uses Type F ATF but you can also use Redline MTL or Amsoil Super Shift. I suggest changing the fluid and overfill it as much as you can. With the car jacked up on one side you should be able to get a bit extra in.
If the shifter feels sloppy, replace the plastic bushings around the pivot ball. The boot comes up, either M6x1.0 bolts w/washers or just push in plastic plugs. The shift knob pulls off, straight up (no set screw, just pull hard). There's a reverse detent plate that guides a plastic bushing to help engage 1st/2nd gear easily, so check alignment of that plate when you put it back together. Then a snap ring you can pop out with a pick & pocket screwdriver. Underneath, there's a set screw on the underside of the shift lever that goes against a pin that locates the shift lever to the selector on the trans. With all that undone, you can remove the shifter from above and dig out the bushing / rubber o-ring front the shifter cage and replace that. I believe you have to also disassemble the shift lever to get the new one on. There's a roll pin through the side of the chrome shifter, drive that out and the chrome part slides up off of the steel lower section.
Those are just a few things that come to mind. Feel free to start a new thread in the Maintenance section with specific questions or even start a Project thread to track and document progress.
-Michael