The whole notion of:
"But, once we make that guess we add it to the price of gasoline and our motorist is once again happy in the knowledge that she has made the world a better place."
...is wrong. They are not "happy." If they were happy to pay the higher price, you would not have had to tax them in the first place!!!
That leaves us with two problems: how can we compensate both the losers in the present (due to higher taxes) and the losers in the future (environment).
We do this by taking the revenue from the tax hike to repay the current losers, and we ASSUME the elasticities are such that carbon consumption falls by enough to provide the adequate benefit to the future.
So you see, the assumption of elasticity IS important if you intend to compensate the current losers (as Mankiw would want to do)
The happiness was a joke. Its a device I somtimes use in class. In economics we assume that people are out to satisfy their own interests but we find that they often end up maximizing social welfare.
I sometimes take the tact that our protagonist wants to maximize social welfare (as many young students imagine themselves) but ends up realizing that all she has to do is act in her own self interest. So long as the government corrects for externalities and the like.
As for the elasticities. You are right that the government does not actually compensate the taxpayers. This does create messy distributional issues.
However, if we were willing to accept Kaldor-Hicks then the elasticity wouldn't matter.
Suppose there were perfect substitutes so that the elasticity was infinte. Then gas would be completed replaced by some alternatives at no cost at all to the consumer. The tax would have the result of completely protecting the environment.
Suppose the elasticity of gas was zero. So that people just had to have it no matter what. Then there would be no change in consumption. Assuming that the damages are not infinte (some might disagree) this is also optimal.
Now, yes we have taken away revenue but we did it a no economic cost. Since, their behavior didn't change there is no deadweight loss. For real nerds, zero elasticity must mean that the compensated elasticity is zero, and that the income elasticity is also zero. If the income elasticity of gas is zero then the gas price elasticity of labor supply must also be zero. This means that high gas prices do not discourage work or investment. In effect we have found the perfect tax.
So at either extreme the gas tax works so long as we ignore distributional issues. By the way those issues are precisely why I think gas tax revenue should go towards increasing the EITC.