Local flight east
Re Mixture Check: My comments are aimed at the procedure for engines with carburators. The point of the exercise is to make sure that the carburator is not running too lean. If the carb is working properly the rich mixture created by the carb heat will be corrected by leaning the aircraft at run up RPM. That is the engine will go from over rich mixture to best power mixture as you lean. This will be indicated by a 20 to 50 RPM rise as you lean. If the RPM immediately drops as you lean then the carb is set too lean. This matters because a lean mixture at high power will create high CHT's and will damage the engine. In extreme cases it may even cause the engine to fail. I have snagged 2 airplanes for this. The first one turned out to have the wrong carb installed
and the second one had a dirty jet.
However since you fly a fuel injected engine the purpose of the mixture check is to check the functioning of the fuel control unit. When you lean you should observe a reduction of fuel flow on the fuel flow gauge and a rise on the EGT gauge. One important check on fuel injected engines is that after full throttle is applied, is to make sure that the fuel flow is right at the max redline or even a little above the redline. Low fuel flows at high power can be caused by a variety of faults and is absolutely deadly for the engine.
Re the Prop Check: Nicks in the prop have sharp edges and the sharp edges concentrate stresses. Cracks then can develop from the stresses and in extreme case lead to blade failure. The good news is props are pretty tough so small dings should be attended to but usually are not a grounding item. Of course if in doubt consult your maintenance organization. A visual inspection is find for checking the prop but touching it is no problem. The issue is moving the blade. Even small movements could cause the engine to fire if the mags are on, particularly with a hot engine. You should always assume the prop is live and if you have to move it never wrap your fingers around the blade. Instead put your fingers tips on the middle of the flat part of the blade, press down and move your hand to move the prop with all body parts clear of the blade path. Before moving the blade make a visual check of the cockpit for mags off, mixture ICO and throttle at idle.
Since we are on the subject of walk arounds, my experience is if I am going to find anything wrong it will probably be a leaking brake, cord showing on a tire or a low nose wheel oleo, so those are things I pay particular attention to.