That can work too, though it's still probably regressive, as some people have more resources to adapt than others.

And if the policy is regressive then you face the danger of it blowing up in your face. For example, see what happened when the rich in America seized most of the benefits of globalisation, now we have Presidents threatening to burn it all down and being popular in doing so. If you actually want carbon reduce toon to be successful you need to factor in the politics, both getting the wealthy and powerful on board and getting the average man in the street to feel that he's not being screwed (which some people will tell him he is, even if he isn't).

Maybe, but I would kind of assume that poorer people already end up emitting a lot less carbon since most carbon emitting activities already have a significant cost involved.

I'm not so sure: poorer people often have to drive further, in clunkier cars. In any event, it's likely more 'regressive' than income: a rich person might earn 10X, but generate 2X the carbon. I bet there's data for this, though.

Because of the refund, 2X the carbon doesn't mean 2X the taxes. Someone who emits 120% of the average amount of carbon pays twice as much as someone who emits 110% of the average, but emits less than 10% more.