What needle depth should I set to prevent the machinery from hitting the needle?
You should be able to download the manual for this machine on Singer’s website. I can’t see the model number, it looks like it may be model 457, if so, it’s there. Upon looking at the manual for the 457, it only gives maintenance instructions not repair. This is not unusual.



See the two lines lower on the shaft?
To answer your “spec” query, instead of a measurement, the instructions will be to line up those marks by turning the flywheel. I don’t remember specifically which line to what spot. 
Then inside the bobbin area, you’ll need to loosen certain screws to move the catch position so it grabs the thread properly and doesn’t smash into the bobbin. 

It seems like it’s that screw, but, it’s the bobbin that needs to be adjusted.  

This would be a “tune up” procedure to fix the “timing” if you want to get quotes from a pro repair company.

Start by thoroughly cleaning out the bobbin area, oil the machine and then, it’s best to get a pro to make the fine adjustments.