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If you look at the downswing of every spike, the temperature sharply decreases and the CO2 lags in decreasing by about 100K years. Which would once again imply that the decrease in temperature eases the mechanism that increase CO2, and that it takes a long time for the CO2 to be removed from the atmosphere. If CO2 causes temperature increases we would see CO2 decrease first even if there was some other initiater of the CO2 warming.
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What are you talking about? There is no lag. I copied and pasted this image to check and the peaks are
exactly at the same location, to the pixel. The same goes for the minimums just before the big peaks.
Each pixel on that graph is about 500-800 years, so we can't say anything about causality (based on this graph alone anyway) here other than that the two are closely linked.
However, we know from basic physics that increasing CO2 concentrations, all else equal, will cause atmospheric temperatures to rise. The physics behind this is almost as simple as the physics behind the fact that increasing solar radiation will cause temperatures to rise.
In order to argue increasing temperatures are causing CO2 to rise you need to come up with a physical process that would cause this. However, any process you could think of will be more indirect and probably considerably weaker than the process that causes temperatures to rise when CO2 increases.