Briggs specifies just one drop of high-quality light oil on the bearing part of the starter spindle. No lube whatever on the balls and in the sprag clutch area: clean those parts like for an operating theatre.

AFAIK there is no model of that carburetor with two holes in the choke pull-out knob. It seems the previous tenant has made an "improvement". You need to put it in the hole shown in the illustration I posted. You might be able to see drill marks in the wrong hole, proving it is home-made.

You do have the idle speed screw, circled in green in this copy of your photo:
[Linked Image]

It doesn't seem to be the correct original idle speed screw, but at least it is there and working. A washer under the screw head would be an improvement on what you have - the correct screw has a bigger head.

Your engine has had some minor alterations made by an enthusiastic amateur, and you will have to find those adulterations and fix them. This often happens, and is a bit irritating, but hopefully you got the mower cheap. Adulterated mowers are worth considerably less than original ones.