This is a tough spot and subject. It is certainly reasonable to think that your driveway should have a surface that is at least in the same plane, meaning now low spots. It may need to tilt one way or another to properly drain but should not have "bird baths" left in it that will hold water and not drain properly.
Unfortunately, outside your contract with this contractor there is nothing that governs this. The generally accepted industry standard would be that the driveway should not have low spots that hold water but getting it corrected can prove to be difficult.
To answer your question, yes you are thinking correctly.
That is correct. He should take a look and correct areas that hold water. No contractor likes to tear out new work but it is the right thing to do. Best of luck getting this taken care of. I hope he chooses to do the right thing. Please let me know if you need any further assistance with this. Thanks, XXXXX XXXXX
Cracks will happen over time no matter what but holding water should not be acceptable.
Thanks, XXXXX XXXXX it!