I have a '73. How would a Cat and O2 work with a 2-v carb?
A cat will burn off unburned fuel in the exhaust.
The O2 will tell you if rich or lean and where in the throttle that happens. The cat takes unburned oxygen in the exhaust and the unburned fuel and burns them. You can't just drive around with a rich fuel mix and have the cat clean it up. There needs to be lean enough conditions to get some oxygen into the cat.
Air injection works a couple different ways. One is blowing air directly into the exhaust ports. The exhaust leaving the port is still hot enough to burn if it only had some oxygen. The air injection adds that oxygen. Still used to this day on some cars as it helps clean up the cold start emissions.
The second part is injecting at the cat. This is used for a couple different reasons. Today it is done so the oxygen sensor does not have a polluted (with air added) reading and can tell what is really happening in the combustion chamber. The old school reason was the cat was a 3-way. The first part was to reduce NOx emissions. That needs a slightly rich (oxygen deprived) condition to break down the NOx. After that reaction it was to reduce the HC and CO, those need oxygen to burn and break down.
The typical exhaust smell people complain about is the HC (hydrocarbon) in the exhaust. Burn that off and the smell goes away. A byproduct is the poison CO goes away as well. NOx is typically ignored unless emissions testing says it is a problem. That is high combustion chamber temps where nitrogen is fused with oxygen, typically in a lean combustion on hot running engine.