Within climate change what you're asking about is called TCRE or "transient climate response to cumulative carbon emissions". E.g. this research article should answer some of your questions. It's an analysis of historical temperature, CO^2 emissions, and the relation between the two: https://royalsocietypublishing.org/doi/full/10.1098/rsta.201...

1. https://en.wikipedia.org/wiki/Transient_climate_response_to_...