Does suffering motivate us to achieve our goals?
In your example you stated that both Tom and Happy had cars that had bad engines.

Tom was unhappy and Happy was okay with it.

So right from the beginning we can see that their happiness did not depend on whether or not they had bad engines but their reaction to it.

Therefore one must apply the same conclusion to the outcome of getting a new car.It's not about who gets the new car first,it's their reaction to it.

Happiness depends on reactions to circumstances.Not circumstances themselves.

Is suffering a motivator?

Yes absolutely.When the Siddhatta saw suffering he decided to pursue the spiritual life.Sometimes living in deva like bliss masks the true nature of reality.That it is impermanent,not self,and suffering.

Is suffering a motivator only if the amount of suffering does not go over a specific threshold?

Yes that is why the human realm is the best place to practice the dhamma.One has their fair share of happiness and suffering.It is balanced to some extent.

If suffering is a motivator, than it does not make sense to completely remove it, because if you remove it, you will be less proactive to reach your goals in life?

That's like saying bleeding will motivate you to bandage yourself.But if you do bandage yourself,then the bleeding will stop,then how are you going to bandage yourself..?? 

Removing suffering is the goal.