It jumping out of gear   could be from not being completly in gear.  Hyundai Tucson 1.6 petrol 201 has had a number of problems with the Hyindia some are listed in Edmonds.   Have the garage check for correct level of transmission fluid.

Have seen this happen where the the temperature blend door actuator goes bad,  Do you hear any noise when you change the temperature settings? 

The  actuator should be on the passengers side on top behind the glove box.   I don’t know if you can see it without taking the glove box out.  If you can see it see if you can hear it  when you change your tempeture settings

On all cars made after 1995, there are 2 types of oxygen sensors,  code PO420 has to do with the exhaust. what is referred to as an “upstream” and “downstream” sensor. The upstream sensor does most of the work in determining the air/fuel ratio of the vehicle, while the downstream sensor is located after your catalytic converter, and it’s there to monitor exactly how your converter is performing

Try some throttle body cleaner,  sometimes a little dirt in the throtle can cause it.

Ok,  If it was me I would verify you have spark.  You said you tried dumping gas in it.  I don’t suggest  for people to use a very little of starting fluid.   Just enough to see if it starts.   If it does I would start by checking the fuel pressure.   With it being a 92 I have seen cars fuel filter clog up from ethenal buildup.

If you are asking about the automatic transmission fluid,  I presume the car is level.  the fluid I believe is lower than the rear housing of the transmission.  With the shaft out and you tilt the car back then you should get transmission fluid coming out the end.

If it started happening little by little I would look at the starter or started solonoid