Quite a bit. Joseph Fourier was first to discover the greenhouse effect in 1824 (
Annales de chimie et de physique). It was carried on further by John Tyndall in 1858 who discovered that CO2 absorbed IR radiation and thus affected temperature on Earth. In 1898 Svante Arrhenius quantified CO2 as a GHG and predicted that it could cause global temperatures to increase by 4 to 5 deg. C per doubling of atmospheric CO2. The modern consensus among climatologists is now 3.5 deg. C per doubling. So, in a nutshell, the physics of CO2 has been well known for well over a century and recognized as a greenhouse gas that can affect global temperatures dramatically.
As you might guess, yes CO2 is the dominating factor in global average temperature. It is well documented in the paleo record and verified with both earth based instrumentation and satellite data. For the previous 600,000 years, CO2 has not exceed 280 ppm nor do proxy records show temperatures exceeding to days current levels. In 1850 atmospheric CO2 was 280 ppm, today it is 390 ppm, right on schedule to a 3.5 deg C doubling (560 ppm) by the end of the century or early next century.
That is not to say that CO2 has been the only factor causing warming, it indeed has not. From 1850 to 1940 increase in total solar irradiation (TSI) can account for about 30% of the warming. From 1970 to present, almost all of the warming is contributed to GHGs, most of which is CO2.
Green line is carbon dioxide levels from ice cores obtained at Law Dome, East Antarctica (CDIAC). Blue line is carbon dioxide levels measured at Mauna Loa, Hawaii (NOAA). Red line is annual global temperature anomaly (GISS).