As for 2, i was assuming there are specific taxes for cars, or at least for petrol. That's the case in my country, don't know how it works in the US.
There are petrol taxes and car registration fees, but that mostly goes to funding road construction and maintenance. This makes sense at first glance, but the article illustrates that it's missing a lot of the costs of driving.
Public education in the US is funded to a large extent by local property taxes. (The result is that rich neighborhoods have considerably better-funded schools than poor neighborhoods.)
Commutes in the US are commonly long enough that it's entirely normal to find people end up driving through poor school districts, while their education-directed dollars go mostly to their own schools. If dealing with the pollution comes from the local district's property taxes (or simply leaves the students breathing polluted air), you've got an uncorrected externality.
1. If the drivers are in a different district, their taxes are primarily going to other schools.
2. The drivers aren't paying taxes to pay for this that bikers, bus-riders, and remote workers don't also pay, so it's not correcting the externality. It's not that the drivers are paying for it; the drivers and everyone else are paying for it.