Only you can decide what you're willing to deal with here. I will say that if you decide to go before the contractor's board, they will expect that you've tried all avenues to correct the problem with the contactor FIRST before resorting to them. And they will look at the specs of the job overall, and what would have been a reasonable result to have been expected given the site conditions and the contract language. If you are the one that set the specs of the job and then got bids for it based on those specs, then you cannot hold a contractor at fault for following the specs that you set forth.

Was the difficulty of installing large format tile discussed with you at all by anyone before you chose it? Even with perfect slab prep, you will have some lippage with large format tile. It bows in the middle. It's the nature of the tile to have lippage when installed in any other pattern than stacked where the bows in the middle align. And that's why manufacturers only recommend a 1/3 offset pattern. It minimizes any lippage issue. Is that the pattern you chose for the tile? Or is it a "random" pattern. That will create more lippage that wouldn't be the contractor's issue. It's the nature of the tile coming in to play. And it would be installed contrary to the manufacturer's recommendations, so it wouldn't be supported as an actionable item.

Any tile floor can only be as flat as the substrate it's installed on. What type of diagnostics did you or the contractor do to determine what type of prep the slab would need? Would you have been willing to pay an additional $500-1K to ensure that you had a problem free install? That would include grinding down high spots, and filling low spots until the floor was flat enough to support large format tile. Was the flatness of the slab discussed at all?

If you would have been willing to pay for better prep, then the contractor should have done that better prep at the additional cost. A true tile pro would have at least brought it to your attention and had you sign off on it if you refused to do so. If the condition of the slab was brought to your attention, then you really don't have a lot of recourse for the issues that you are seeing.

As far as the cleanliness of the job site, contractors usually stipulate "broom clean" which doesn't mean move in ready. It means the majority of trash is gathered up and the place is swept. It doesn't mean mopped and scrubbed. You should always plan on cleaning a space post renovation, or hiring a cleaning service to address the job. Renovation produces lots of dust and dirt and it gets everywhere. It's not your contractor's job to deal with that unless it's written into the contract that he will hire the cleaning service upon job completion. That would be a higher bid for the job, of course. It sounds as though your tile job simply needs a good initial scrubdown.

The tiles have all been scrubbed down and the grout water marks or whatever was there are all gone. The grout lines however still appear very light / white, and to be honest, I have not looked into how to clean that yet, nor tried (other than a little soap and water).

you present a lot of great questions, which I realize now, but at the time had no clue of.

First time home buyer. No idea about contractor work or anything along those lines. I just knew to hire someone licensed and not a random craigslist person for cheap.

The only specs I set for the job were the type of tile to be used (as mentioned) and how it was to be laid (which he suggested brick style and I said no 1/3 like the box says).

I had read large format tile was difficult to install on DIY type boards, so I assumed thats why people would not DIY it. Hire a pro. They know.

best question:
"What type of diagnostics did you or the contractor do to determine what type of prep the slab would need?"

I have no clue. I called him. He came over. Looked at the house. Measured the rooms. I told him the rooms I'd remove carpet from. Told him the rooms I wanted him to remove existing tile from (kitchen, bathrooms), and told him I wanted him to remove the laminate flooring in the entry as well as baseboards throughout the house.

I dont think any of that falls under the category of diagnostics, but again, I'm not a contractor.

another 500 - 1k would have been fine if the foundation needed some leveling, touch ups, etc.

no slab flatness discussed until I visually saw the two tiles that needed to be fixed. Then I was told the slab was not even there. ok, well neither are the tiles so fix them please.

cleanliness was not that big of a deal. but man, what a mess it made of my garage.