I was wondering which kind of plants would survive longest without CO2 : plants with fatty, thick leaves or plants with thinner, less fatty leaves? And can we conclude that the plant species that are more dependent on CO2 levels, would benefit from the greenhouse effect?
Plants do not need CO2 for survival. They need it just for photosynthesis; if you supply them with alternate carbon source such as glucose then they would survive (See here for non-CO2 Carbon-source utilization). There are non-photosynthetic plants which are totally dependent on carbon sources other than CO2; some are carnivorous like venus fly-trap.
I am not sure what you intend when you correlate leaf thickness to CO2-dependence. Leaf morphology can affect photosynthesis rates but afaik there is no relationship between leaf morphology and CO2-dependence.
can we conclude that the plant species that are more dependent on CO2 levels, would benefit from the greenhouse effect
Greenhouse effect is not equivalent to increase in CO2; increased CO2 levels can cause greenhouse effect which is a physical phenomenon of entrapment of IR radiation.
CO2 is fixed by an enzyme called Rubisco; like most enzyme catalyzed reactions this will reach a saturation limit i.e. addition of more CO2 will not increase the rate of photosynthesis. However the rate of photosynthesis is limited at the Rubisco step because of low levels of CO2 (See here) and it can be expected that most photosynthetic plants will be benefited by increase in CO2. However, not all plants would be resistant to increased temperatures and moreover the rate of enzyme catalysis would also change with temperature. So it is not that easy to predict.