AFAIK Canada does this by giving everyone a fixed amount of tax credit to offset the tax. That way poor people aren't taxed more but everyone are still incentivized to reduce their emissions
That way, if you use less than the average amount of carbon, you actually make money.
2. whether they pay or not is irrelevant. what actually matters is whether they end up paying more or less in net. for a scheme that taxes people according to their consumption but gives everyone the same amount of tax credits in return, poor people would be expected to come out ahead because they consume less on average.