Temperature differences y/y are very small compared to natural
variation during the year, and the trend we see could be statistical
illusion, especially that the scientists who developed the method were
looking to demonstrate exactly that sort of trend.
Fortunately statisticians have developed tests to determine whether the evidence for a warming trend is "statistically significant" (i.e. the probability of observing a trend as great as that seen if it was just a statistical illusion is below some pre-determined threshold). This is basic STATS 101 stuff. If you pick a trend length above the WMO recommended minimum of 30 years (i.e. a period long enough to expect the trend to be detectable given the noise - in other words the test has reasonable statistical power) then the evidence for a warming trend is indeed statistically significant.
The correlation between CO2 emissions and measured temperature changes
is not perfect.
Well of course not, because CO2 is not the only thing that influences global surface temperatures, for instance volcanic forcing, aerosols, solar forcing etc. You would only expect a "perfect" correlation if CO2 were the only thing affecting GMSTs and there was no "weather noise", for instance ENSO changing the distribution of heat between the atmosphere and oceans.
Ivar has been rather naughty in his diagram, you can get a better visual correlation by rescaling the graphs:
Sadly woodfortrees.org only has the Mauna Loa data, but you get the idea, if someone is claiming that the correlation isn't very good and don't plot the data in a way that maximises the apparent correlation, then that should ring alarm bells!
I suspect the correlation with total heat content (dominated by ocean heat content) is likely to be rather better (as increased radiative forcing heats the oceans as well as the atmosphere, so you need to consider them both together).