The Vostok core picture above shows that is impossible in this case. If the CO2 rise were natural and caused by warming, the world would have had to warm 10C on average since the Little Ice Age in order to produce the 280 to 400 rise. In order to produce the rise of about 25 ppm per decade, the oceans would have to be warming 2C per decade at least. That is simply not the case.
Hmmm0- the hockey stick graph?
Yes, a type of hockey stick, but accurate in this case. The ice core CO2 measurements are very crude by having only one CO2 measurement every few centuries (in the oldest cores) and having diffusion burring that measurement. So there could well be prior hockey sticks in CO2 that cannot be visible due to those factors. But there is no other evidence for CO2 hockey sticks in the past, not in the low resolution cores, and not in the high res cores from Greenland going back to the last ice age. The CO2 hockey stick is unique.
It is quite true that CO2 rises follow temperature rises, nobody disputes that. What is also true is that a temperature rise of 1C produced about a 10ppm CO2 rise some number of centuries later. But we have a 120ppm rise and still going up 2-3 ppm per year.
Hmmm, that is contrary to global warming alarmists claims
Quite true. Climate alarmists mostly do not understand the way earth's temperature is regulated. Greenhouse gas gives us a very rough range of temperature but the actual temperature is determined by weather. The simple fact that the earth's average temperature can rise or fall 0.1 or 0.2C in a couple of weeks should make that obvious. That is the equivalent of decades of CO2 rises theoretical contribution to temperature.
Heat rises, as heat rises, it cools, and unless CO2 is capable of acting as some sort of massive super furnace mechanism,
You are mixing up convection and radiation. Air that has been heated by the sun at the surface rises. That is convection. CO2 and other greenhouse gasses do not hinder convection in any way. In fact an increase in convection can easily offset an increase in GHG.
Radiation specifically infrared is what is hindered by CO2, water vapor, clouds, etc. Take a look at this satellite photo: 
which is false color, but the clear areas are warm (lots of outgoing infrared) and colored areas are less outgoing IR. The blues are cold cloud tops, hence little outgoing IR, one of the many ways that earth's temperature is regulated by weather. This IR measurement has decreased on average over the years:
but it is highly seasonal and very variable based on the average of global weather. This is yet more evidence of the relatively small role of CO2 in setting the global temperature. The best explanation for the decrease in OLR over the years is increased CO2. The specific spectral components also point to CO2 along with increased H2O, the latter can be completely natural.
The rest of your post mixes up convection (heated air rising) with radiation (heat lost to space at the speed of light). It is the case that radiation in certain wavelengths is absorbed by CO2, H2O and other GHG. Those newly warmed molecules nearly instantly transfer their heat to the O2 and N2 surrounding them. When there is more CO2 and H2O there is more capture and more warming. Those same molecules also release extra heat in the exact same quantities that they absorb it. The thing is, they release it both up and down so some goes back down to the surface.