Battery cables got cut a few days ago as someone wanted my battery. I am replacing all cables but am trying to figure out how to get the positive cable off that goes to the fuse and relay center, and trying to trace the positive cable that goes to the starter and to alternator? Best way to tackle this problem?

The red one goes to the positive. the other one goes to Negative on the battery.

The key line was bad,
if you had hot wired any starter solenoid on that tiny pin
then in 1 second you saw it cranks ok, ending 2 starter pulls.

then saw it green with corrosion, or cramp bad bad or lug bad.
they sell new FASTON(tm) crimp lugs in stores with the correct tool to do that.
you can make a new wire, and crimp it sitting at a table.
then solder and heat shrink it back to the harness wire.
this is like electrics 101 here.
its just a wire. and a lug,.
or do that all laying on your back under the car,
your call.
all this is very easy, why not ask any shop to do that.

To test a altinator the easy way without special tools, start the car, remove the negative battery cable. If it stays running, the altinator is good. I had the same type of problems. It's probably your ignition coil/coils going bad. Try replacing it and cleaning inside distributer cap & rotor.

there is a chance that it is a battery backup alarm, usually run by a few batteries.

P0123 is a TPS/Pedal Position Sensor A Circuit High Input code. P0162is an 02 Sensor Circuit Malfunctoin (Bank 2 Sensor 3) problem.

Suspect the positive battery terminal is either corroded, or the cable itself has corroded through. If you have a meter or test light, look for voltage at the starter solenoid where the positive battery cable connects to. Have someone try the starter while watching this point. If the voltage disappears during the start attempt, try replacing the positive battery cable. There are replacement ends available (for the battery end), but they are problematic at best. Just remove the old terminal, cut the insulation back somewhat, and mount the replacement on the cable (two small bolts). Hope this helps!

Bad alternator.

You may have corrosion on the battery terminals. Remove the negative cable first, then the positive. With a special wire brush (made for the purpose and fairly cheap), clean the connector interiors and the post exteriors. If the connector bolts are badly corroded, you can find spares at the auto parts supply where you got the cleaning brush. Apply anti-corrosion grease (from the parts house) to both posts and terminal interiors. Reinstall the positive terminal first, then the negative. Be sure to wash your hands after this to prevent mysterious holes in your clothes. If the battery is discharged, recharge it overnight with a low amperage charger. Take the car to the parts house and have them check the operation of the alternator after you get it running. Hope this helps.

CHECK FUEL PUMP REGULATOR