Hello Adro
Firstly welcome to the forums
I have added the photos for clarity/to help future patrons
As you have identified, you have one of the last 158cc briggs L head (flat head side valve) engines.
Though yours is made in china, they are amongst the hardiest engines ever made and what they lack in refinement, they make up for in longevity
Firstly, you have been sold the incorrect governor spring. The one you require is this one
https://www.gaspares.com.au/products/GVS8002It fits through holes as per my photo annotation
The RPM to set it at really depends on your lawn care habits. Since I regularly mow and prefer to leave the mower running idle (yes 'dangerous') whilst emptying catcher, I tend to set mine to 3000 rpm so I yield a lower idle speed for warming the engine up and so I don't get a face full of grass when removing catcher.
You can go up to around 3300rpm (they came from factory at about 3150) but I wouldn't suggest going any higher as the alloy conrod won't like it long term
Plug is a Champion RJ19lm or NGK BR2LM at 30 thou inch plug gap.
This engine has no auto choke - it relies on primer to start
If you mean how to adjust the rpm, it is adjusted by bending the tang forwards or backwards (as per annotation) to slow the rpm by reducing spring tension, or tightening spring tension. DO NOT adjust by bending the spring.
I would personally pull the carby again, clean everything (particularly mesh screen on the tube going down into tank) with new diaphragm and gasket
Your fuel tank paint is flaking off - which indicates to me you have a small leak between the carby and tank.
If you save the old gasket and put it back together (from top to bottom) old gasket, new gasket, diaphragm, and tighten the carby as per this briggs illustration, you should be fine.
Do not torque them down like a head bolt, just tight with a phillips screwdriver
There are clear signs of dirt ingestion into your carby.
Please clean the air filter in hot soapy water, then dry and re oil by squeezing/rubbing 2 table spoons of oil (preferably chainsaw bar oil if you have it) thoroughly through the filter, then blot out excess with a rag.
If you do not reoil, the filter will do practically nothing
lastly, the model number is actually (0) 10T802 - which is why you got no results
Hope this helps
Regards