Only my opinion, and only based on the verbage of the contract image you provided. . so take it as you will.
I did not see anything in the contract with regards to grading and re-compacting the subbase material immediately below the existing asphalt. I would have the contractor detail something in the contract with regards to grading and re-compacting. You want to do this prior to paving.
A firm subbase is a major factor in pavement longevity. Your existing pavement, as you said, is cracked and there are low spot puddles. These two things could be letting water into the subbase, resulting in the subbase holding water. .which contributes to soft areas, thus accelerating your new pavements deterioration.
Please have the contractor clarify if the pavement thicknesses he provided are after compaction of the mix, or before. You want those thicknesses AFTER compaction.
Also, I would confirm that the contractor utilizes the vibratory function of the compaction equipment. Saying you will. . And doing it can be two different things. On most rollers. . It's a simple switch that turns the vibes on and off.. . And my experience tells me that most operators don't enjoy rolling with the vibes on. If you aren't watching, they aren't vibrating.
Achieving proper compaction is the single most important factor when it comes to pavement durability and longevity.
Another side note, please do not buy into the contractors claim that they are using "Airport" grade mix. My opinion is that this claim is only a gimmick. It is extremely highly unlikely that you will be getting actual FAA P401 asphalt mix. FAA airport mixes have substantially different aggregate gradations, and liquid asphalt grade when compared to standard commercial, and DOT mixes. These factors increase the per ton cost. And there's no way an asphalt producer is going to run 5 or 6 loads of FAA P401 "airport mix" for a driveway guy. The cost benefit just isn't there for the producer.
More than likely you will be getting the commercial binder and top mix designs that are hopefully are coming from a DOT approved plant, using DOT approved materials. The only way he's getting airport mix. . Is if it's already been rejected from an FAA job that happens to be going on the same day your driveway is getting done.