P0327 is a knock sensor code. P0056 is a HO2 sensor code. P0161 heated O2 sensor code. P0562 ignition voltage low code. P0700 is a transmission fault code. So with that said, I would primarily focus on P0562 and P0700. As derf said you will need a special tool to get a better idea what exact codes the transmission control module has. P0700 is a generic code that says that TCM (transmission control module) has sent a request for MIL (malfunction indicator lamp) to ECM (engine control module). Now if you car was running fine and that code appeared, I could have said that either TCM or ECM is bad as possibility. But, since you said your car is having issues, with that code set I think there is more that is going on with transmission than the codes you gave us. As far as heated 02 sensor codes (HO2S), if you diagnose the P0562 code, the others might get fixed too. P0562 is ignition low voltage code which says that your ignition system has lower than minimum required voltage to run the vehicle. The codes for HO2S is all have to deal with circuit voltage low, so all of those could be related to same thing.

 

I concur. In fact, the P0562 is likely at the heart of this.

The code sets when the system voltage is lower than 10V for 60 sec. It should be in the neighborhood of 14.4V. Not only will this cause other sensors to generate crap for data, throwing false codes, it may very cause the transmission control module to misbehave. And the solenoids involved in shifting gears are likely in chaos.

My first move would be to get a full load test done on your charging system. This is available for free at AutoZone. Ask them to run it twice.
Your alternator supplies current to all the electronics in the car AFTER the car is started. If the requested current cannot be satisfied by the alternator, I suppose in theory it could pull from the battery but if so, then of the current requested is met, it should be able to do so at 14.4V.

But it isn't.

I believe Saturn started using ECM controlled alternators to vary the output current of the alternator. If so, there should be a 3 or 4 wire lead connected to the alternator that runs back to the ecm. Wiring problems here will likely make for an unstable mess.

So will poor connections at the alternator output, it's case grounding, both battery terminals, and the - batt ground connection.

Those are easy things to remove clean and reconnect.

My bet is that the voltage regulator inside the alternator is flaking out when it gets hot. This can probably be verified by reading DCV at the alt output vs chassis ground. If the voltage is unstable, the v reg is likely to blame.
Do this measurement right after startup and then when everything goes whack. You will likely see a stable V at cold start-up (pls record) and an unstable one when things go awry. It is not at all uncommon for electronic components that are marginal to act up and begin to fail when they get hot. I'm thinking that if it is a voltage regulator failure problem then it may only show if the load test is done while the car is acting up but I cannot be sure of that. So I would take it for the load test after it has begun acting up.

If after testing, the alternator is at fault either through not functioning properly or the voltage regulator being bad, I would recommend replacing it with an oem part. The computer-controlled alternators can be very finicky as we saw with a post last week from someone with a 2003 Vue. You do not have to get an oem which I believe would be AC Delco but it is my recommendation. It will be more expensive. as for remanufactured versus new that is up to you. Reman should be fine for your purposes.
There is also a chance that there is a bad cell in your battery that is either shorted or simply cannot store charge. This would cause the overall voltage to be dragged down. However, because you have not mentioned issues with starting the vehicle, I do not believe this to be the cause. Just mentioning for the sake of completeness.
There is also a chance that it could be the ECM, but let's not go there until we have to.

So don't throw Parts at it. Go and get the free load test. It should give you a lot of information. Please post the results.