Short answers, pulled from the story.
Scientists define climate sensitivity as the change in Earth's surface temperature that results from a doubling of atmospheric carbon dioxide concentration. This measure helps researchers understand how much warming will occur due to radiative forcing starting from 1750.
Svante Arrhenius published the first paper quantifying global warming consequences in the 19th century. He initially estimated a rise of around five degrees if concentration doubled, later revising it to two degrees.
Reaching this state can take centuries or even millennia because deep oceans act as heat sinks. The ocean takes many centuries to reach a new steady state after such perturbations.
A committee chaired by Jule Charney convened in 1979 and estimated a range of 1.5 to 4.5 degrees Celsius plus or minus one degree. This assessment followed earlier calculations performed by Syukuro Manabe and Richard Wetherald in 1967.
The uncertainty in estimates mostly stems from how complex interactions play out over time including cloud processes and water vapour changes. Higher temperatures cause ice to melt which reduces sunlight reflection and traps more heat energy while balancing feedbacks tend to increase the rate at which energy radiates back into space.