It doesn't tell us what it should look like. It doesn't tell us how much each person should pay. It doesn't tell us how much each person should be compensated.

The fuck it doesn't. To find the amount to pay, add together the quantifiable costs born by others as a result of the externality and divide it by the amount of good/service that produced that level of externality. $X per metric ton of (insert pollution here) released. The great thing is instead of trying to determine who's compensated what, you can use the new revenue to eliminate other taxes or direct funding to counter the effects of the externality. So even in a scenario where the tax isn't perfectly set, it still results in a much smaller market distortion and less economic dead weight loss then the option of no interference.

It tells us what the Pareto effecient level of production is if we know what people's preferences are, which we don't. If people don't have quasilinear preferences, or if we don't know what people's preferences are, economics doesn't tell us what to do.

Not incurring unnecessary and otherwise avoidable healthcare costs, upper respiratory illnesses, or premature death as a result of someone else's actions without your permission is a pretty universal preference.

That's the textbook example. However, if you look at it from a libertarian perspective, it is not clear that a government's military is a good. It is not clear why your government would rule you better than your government's enemies.

Which proves your point completely moot as you then either A. live under a government you support and are willing to pay, B. Live under a government you support and just want to free ride on the support provided by others, or C. where you live under a shit government you can't leave and whose enemies' government you prefer in which case your government really doesn't give a fuck and is going to take it anyway.

I'm not convinced that is true. I know many cases where it certainly isn't, where the cost of a public project is inflated to astronomical levels.

Private enterprises have cost overruns as well for large projects, but it doesn't matter in the end as the cost is passed onto the consumer who willingly pays it. If the cost goes up too high the customer won't pay later when the cost is absorbed into the final price. These inflated cost projects are always run by government agencies supported by a general fund and not government, partially government owned, or private partnered, corporations supported by the revenue/tolls/fare collected.

I don't really have a comment about your road building paragraph. Maybe there are better ways to build roads with a government. But it's entirely possible to build them without one.

It is possible...just not without creating a monopoly wherever it extends for everyone on/traveling on it. Whether it be just that street or a monopoly running through an entire city. For building a road that's great and exactly what I said before, but granting full private property rights complete with excludable power on common carrier infrastructure which can only exist in the form of a monopoly without mitigating regulations and oversight to prevent abuse of natural monopoly is to ignore even an ancap's praxing of market incentives.

I would argue that such a thing would be an improvement. But it could still be improved further by eventually getting rid of it completely.

Negative externalities violate the NAP. I don't understand your acceptance of allowing property and violent crime to be committed against someone so long as it is just an obvious and unavoidable side effect of their intended action.

But how much of it? Most goods aren't binary. It's usually not the case that you either have it, or you don't. And what should the payment schedule look like?

...As I already stated, how much the government/company spends = capital, operating, maintenance costs of that route, and that route exists so long as the tolls, fares, revenue supports those costs at the bare minimum. Those tolls, fares, and revenues can only be collected if people want to use the good or service in question. This is a basic explanation of how markets work. The reason the government is involved at all is to prevent the market failures unique to this situation.

So it has nothing whatsoever to do with whether or not people actually like the good or service?

Ok now you have to be deliberately misinterpreting me to get that. Since the good or service is paid for via toll, fares, or consumption tax on those who use it, it can only continue to function by attracting and keeping users otherwise they stop using it and Ta Da! No more money to run it.

Please look up how the MTR works in Hong Kong.