Ping noise when turning wheel and screech noise when braking

The ping is probably a binding upper strut mount. That has to get really bad before it becomes a safety issue. I have never seen one so bad that you cannot steer the car but it could prevent the steering wheel from coming back to center on its own after you turn a corner. Long before that you would notice an irritating need to constantly make little corrections. Most of the time all you will notice is the noise.

Reach over a front tire and lightly wrap your fingertips around one of the upper coils of the coil spring, then have a helper slowly turn the steering wheel. You should feel the spring rotate smoothly with the wheel and tire. If the upper mount is binding, you will feel tension build in the spring, then it will pop loose and continue to turn.

The mount needs to have the car's weight on it to bind like that. Very often that kind of wear does not show up when the strut assembly is taken apart. There are other types of wear that can occur that can't be seen until they are disassembled to replace the struts, then the mechanic has to tell you more parts are needed than just the struts. They do not like having to do that so many shops just include new mounts in the estimate when they quote you a price for new struts.

For the brake noise we would have to know more about the brake service. If proper procedures were followed, the mechanic would have machined the rotors. In that case the most likely cause of screeching would be the quality of the new pads. Higher-quality pads are relatively hard and tend to squeal, especially in hot or humid weather. The clue is that squealing goes away when they get hot from city driving. You may hear a subtle, rough grinding noise at slow speeds too when you are not applying the brakes.

If the rotors were not machined or replaced, there will be ridges of rust where the old linings did not ride. New linings are never exactly the same as the old ones and will ride on that rust until it wears down. That will cause a rough sound. Not machining the rotors is not the professional way of doing a brake job. It is how some do-it-yourselfers replace brakes.

If your pads use some type of anti-rattle springs or clips, one of those could be mispositioned or bent and is rubbing on the rotor. Typically that noise will occur whether you are braking or not. Take the car back to the shop that did the work and give them the chance to check their work.

Thank you so much for answering my questions Caradiodoc! That has been the most comprehensive answer I got from anyone.