That matters very little. Carbon taxation is a Pigovian tax, and just making emitting CO2 more expensive creates strong incentives to stop emitting CO2. It also makes "green" alternatives comparatively cheaper.
In fact, many proponents of carbon taxation support equal redistribution of the tax money to everyone (e.g. as tax credits). Taxpayers polluting less than average would see their tax burden decrease, while those polluting more than average would be paying more and thus be incentivized to change their behavior.
(Of course, it's not that easy in the real world, some behavior can be extremely hard to change without help/subventions/large investments. But these helps don't have to be tied to carbon taxation.)
Well that would drive up demand for fossil fuels and profits for extraction companies, leading to more extraction. Is this a real idea (proposed policy in a country) or just something you wrote down?
To be fair, many small/mid-sized companies that act as suppliers to larger companies would basically have the larger company doing the carbon accounting for them.
For large companies, I think it would look more like an accounting capability than a climate/energy science one. The carbon credit market might become regulated with some accreditation system after these wild west years.
Start first with usage of private jets or TV sets?