Will my car turn better if I install aftermarket spoiler?

I don't think so. Even with no visual upgrades I can drive similar times.

In reality, yes. But since it's arcade-style game, putting a spoiler does NOT affect the performance stats.

Would adding a Spoiler to the car or any car that you use in the game affect your steering and get less times or do you have to turn off Stability Control for better steering? I want to get better on the turns.

Adding a Spoiler has no affect.
You can try to turn off Stability Control and see if this works better for you. There is definetly a difference between Stab on and off, but you can still be fast with both. It sometimes depends which track, or which controller you are using, if stab on or off is better (and personal preference).