" It’s a mess and we need advice on what to do from here. "

Get another roofer to look at it, asap. But as stated above, wrong product for that application. Hopefully you have not paid this guy.

It should have at least a rubberized membrane underneath.

You don't put roll over rubber.

" It should have at least a rubberized membrane underneath. "

You can't put roll over that and it would defeat the purpose when you nail down the roll roofing. Roll roofing is for a minimum of 3 1/2 / 12 [itch roof. That is dead flat and needs a commercial product. Nothing shy of ripping and doing it over will fix it. Water sitting on that will always find it's way in.

That looks like modified bitumen which has been used on commercial buildings for a long time. I've speced and contracted dozens of flat roofs but can only tell the difference between MB and 30 pound roll roof membrane if I'm standing on it. Not sure how the above folks can tell from a picture so since I have always been and remain a student I'll ask. How do you all know that is roll roofing and not MB ? Even if it is approved material it wasn't properly installed and the ONLY remedy is remove it and start over. I lay 10 to 1 odds the installer isn't insured nor bonded. I lay even odds some if not all installers were working in a lettuce field in Central America a week before getting on your roof so your chance of collecting damages in a lawsuit are zero. For your sake I hope I'm wrong because this could turn out to be a very expensive lesson on vetting contractors before doing business with them. Which brings me to the question of why are you on the net looking for answers instead of pursuing the contractor?

"We had several nails driven through the ceiling in the flat roof area by the workers."

Nails mentioned by the OP, and it looks to be roll roofing by the edges, looks like asphalt paper. Maybe making some assumptions but like you said, tear off and re-do by someone with knowledge is the only answer.

There should never be ponding on a proper roof. That's a tear-off do-over. I'd suggest a reputable EPDM contractor.

I was actually waiting for another contractor to come and take a look as I made this post. I am a student as well and this is not my area of expertise. I do appreciate your honesty. I had someone from his company come this morning as well to assess the damage done. He seemed shocked by the extent of the damage and said he hadn’t seen anything like it in 10 years. They used Topsheild SA Cap so it is a modified Bitumen. For clarification, the nails coming through the ceiling were from the drywall. The owner called again this afternoon. He said he will do whatever it takes to make it right. I did tell him that I am leary of the work at this point.

I have a couple more roofers coming to give second and third opinions. Before he saw the damage they were going to have guys come and seal it better. I said absolutely not, there is clearly water underneath and I was concerned about mold. I told the gentleman this morning the roof needed to be removed as it was clearly not the correct product and there was a great deal of moisture underneath. Would TPO be the correct product and would it be ideal to slope it so it comes off the roof?

Something similar recently happened to a friend. She immediately contacted her insurance company and they took charge, which has been a blessing. They are dealing with the contractor directly to make it right, but also sent in a remediation company that installed dehumidifiers, floor protection, etc. and will repair the interior and exterior damage once the roof is properly repaired. Call your insurance agent ASAP!

I called our insurance agent just to keep them in the know about what is happening and they said this should be covered by the roofing companies workmanship insurance and that they are not responsible for this.

Stop looking for free rides ,hire a qualified roofer or consultant to speak for you. When it's over you will realize the money was well spent,saved your nerves,saved money and resulted in better job.

TPO would be a much better choice (if done correctly!) and yes building up the deck with sloped foam to create pitch would help as well. They need to get a remediation company in here asap to dry out the interior.

"– Address drainage deficiencies to provide positive drainage;" Page 12.

Update and additional advice needed-

I had two seperate roofing companies take a look and give a 2nd and 3rd opinions . 2nd opinion recommended modified butimen torch down. Said it would be appropriate for West Texas weather. Did say it was not for ponding of more than a quarter inch ( we have areas that have quite a bit more than that). Did say it could be built up in these areas.
-3rd opinion recommended basic flat roof TPO installation. Said original install was wrong all the way around. Not the correct application, wrong drip edge, vent pipes were also not for a flat roof etc. said he thought TPO could be installed over peel and stick with a layer between. Recommend screw down.
-original contractor came and looked at roof as well. He also recommended basic TPO installation and said he would take care of it and make sure it was right. We did asking him about a graded TPO interlocking system to try to avoid ponding water. He had a company draw up a plan. Is it worth it to do the grade or should we just go with the basic TPO?

https://www.dropbox.com/scl/fi/yqntwm69wgmz84gufayq7/232-03927-S022443-1-Drawing.pdf?rlkey=ivdr7dozjcjkqwq8hbwyx2l5d&dl=0

The TPO manufacturer tells you specifically to "– Address drainage deficiencies to provide positive drainage;" as linked above. How much more clear can they be?

Pretty long run on the one side from the picture. You would be looking at a bunch of tapered ISO and bringing it way up the roof if you go that route.