So a carbon tax on electricity generation sources works ok to incentivize a switch and internalize the externalities, but it's pretty equivalent to the general ITC/PTC subsidies that have recently been passed, with the exception that they are sector specific.
An economy wide carbon tax does almost nothing to incentivize decarbonization of steel or concrete, as the tech isn't there, nobody has spare money to invest in a solution, and the tax wouldn't necessarily increase the price of end products much. They need very specific industrial policy in these sectors to develop alternatives and put them into production.
Finally a carbon tax creates an entirely unified political coalition, across many sectors of the economy, to oppose the tax. So setting the tax at a level that is right to incentivize change is guaranteed to be nearly impossible, because od the breadth and political strength of the opposition to it. Specific carrots and sticks for each sector are what can drive real change in each sector.
If it was 1980, I'd be supporting a carbon tax. But we need to move far more quickly now to change our economic production systems than a carbon tax could ever conceivably achieve.
I can't think of a case where exempting some group from a carbon tax would be good policy. There is a pretty good argument though that a carbon tax would be regressive, and probably should be compensated for in some way so that poor people don't come out worse off, such as by reducing income tax in the lower brackets.
I think the hardest sector to decarbonize is going to be the military (with the exception of some small percentage of navy ships that are nuclear powered). I could see plug-in hybrid army vehicles eventually being a thing, though. Being able to run off of either diesel or electricity provides a lot of logistical flexibility. You can save the petroleum for when you really need to move long distances.
I'm generally anti-carbon tax because it's the solution to the problem that is obvious, simple, and wrong. But when there are no other solutions, charging for cleanup is the way to go.
Farmers and miners pay for their costs. They can continue to do so and pass their costs on. Some farmers, at least in the US, we massively subsidize to a ridiculous extent in a make-work program for massive amounts of soy and corn. But this is an economic choice that is relatively recent in the history of the US, and we can decide at any moment to improve the efficiency of the economy by making externalized costs internal.