Having looked through the paper, I have some comments on it.
I don't see how the insurance study could have revealed the intended effect even if it had been run perfectly (and this leads me to a guess as to why it was falsified). The customers reported their odometer readings and were supposedly falsifying them in order to get cheaper insurance by appearing to drive less distance. This is not plausible and would have easily observed consequences. The paper mentions that the average mileage in the US is about 12,500. So imagine a customer buys a two year old car which has done 25,000 miles. They buy insurance, reporting the correct mileage. Next year they want to appear to do lower mileage than they really do, so they don't report 37,500, but 35,000. All is ok for now. But next year, the true reading is 50,000. If they deduct the same fixed amount - 2,500 - then they merely pay the correct premium as it appears they have done 12,500 miles and they really have. To get the same advantage this year, they need to deduct 5,000 miles and report 45,000. And so on. Every subsequent year they have to calculate the correct amount to deduct and report a figure which keeps diverging from the true amount. If this was really happening, the insurance company would be delighted! When such a customer had a significant crash and a report on the vehicle was produced to assess whether it's economical to repair, it would show that the odometer reading was sigificantly wrong - and more so the older the car was. This would mean the fraudulent application for insurance was exposed, so the insurance company could refuse to pay out. Insurance companies are always looking for ways to avoid paying claims, so it would become well known if this was happening.
However, if we assume that when the study was being designed the people designing it assumed that customers would report their annual mileage directly, rather than odometer readings, then it starts to make sense. A customer could plausibly deduct 10% every year and if they were challenged that these didn't add up to the value of an odometer in a crashed car, they could plausibly claim that they were only off by a small amount and it was an honest mistake (especially as few people keep meticulous records of their mileage). Presumably, an insurance company was asked to modify their forms for the experiment, but nobody looked at the forms before they were sent out to see how mileage was reported. Once the data had been gathered, somebody realised that it was all useless as it showed the current odometer reading instead of the estimated annual mileage. We can't tell who this was - maybe Dan (as all the authors agree he supplied them with the data) or someone at the insurance company. But they sought to salvage the situation by inventing fictitious second readings.
Another, minor point with regard to the other studies is that it seems odd that having no signature always has less cheating than signature at the bottom. If a signature at the bottom really does have no effect, you'd expect it to appear randomly better or worse than no signature. Maybe it's just chance, since the number of cases is so small.
There's also a problem with the other experiments (apart from the laughably implausible idea that what changes people's behaviour over $1 will change their behaviour when $10,000 or $100,000 is at stake). The paper says that the participants were paid a "$2 show-up fee" and, at the end of their puzzle task, filled out a (fake) tax form to claim payment. However, there's nowhere on the form to put the $2. Surely participants would have noticed that and questioned it. Why didn't the experimenters either include it or mention the mistake in the paper?
And there is a slight chance of a very nasty sting in the tail. In some juristdictions it is tax fraud to falsely tell someone you are deducting tax. It's not clear whether the particpants were told the truth before they went home.
From the paper:
In court cases, witnesses verbally declare their pledge to honesty
before giving their testimonies—not after, perhaps for a reason.
Obviously the reason is to get them to tell the truth. Part of this is by getting them to behave ethically and part by putting them at risk of getting convicted of perjury if they lie, and for the latter you want people to be really clear on when the period start that makes them liable. When filling out a form, there is no possible doubt over when you start doing it, but you're free to make mistakes, cross things out, or simply start over if you feel you need to. Then the signature means that you are finally satisfied that it is all correct. This inherently means you must sign as the last thing you do. Of course that doesn't prevent the physical signature being at any particular place on the form, but its natural place is at the end. If there is a valid phenomenon as described in this paper, I would expect you'd need a signature at both start and end to exploit it while also catching mistakes by triggering people to review the form just before they sign.