I think you're on the right track. I'll leave what technical questions to ask to our builder experts. The question that I would ask of any involved (PB and engineer) is for proof of insurance. It'll be in the form of some sort of certificate that you can verify yourself. Any written warranties or engineering stamps are worthless if there is not some deep pockets involved, and that means liability insurance. Not just a license or bond, actual liability insurance that covers all the things that can go wrong.
You should have a valid contract from each party involved. And you need to read that contract, especially any clauses that give them an out. We had one of those posted just the other day: contract specified all the work to be done just fine, but had a couple of clauses that basically said "We're not responsible for anything that goes wrong."
You need to approach this from both sides. You've been asking questions and learning about each side, but they are not mutually exclusive. You need a crew that can build this pool correctly given the existing conditions AND you need real assurances that they will stand behind that work should something fail. The former is provided by an engineer and a PB that can follow his plans, the latter is provided by the engineer's stamp and verifiable liability insurance of sufficient coverage (ideally for all vendors, not just the PB, not just the engineer).
The knucklehead that DQ'd himself, the one that just wanted to dig a deeper hole and fill it with gravel, did you a favor.
If they can build a city of high-rises on landfill, they can build you a pool in your backyard...