It is said that carbon dioxide levels were much higher during the Cretaceous and Eocene periods: which proxies are used to determine paleo-pCO2?
It seems like you are asking about proxies that are useful on the scale of tens to hundreds of millions of years. We have great proxies that going back thousands of years (tree rings, peat cores, lake sediments), and quite a few that go back around a million or so (speleothem, ice cores), but beyond that the data is in sediments and rocks.
Deep sea sediment cores are probably the most important source of proxy material going back tens of millions of years, and carbonates the most useful lithology for retrieving proxy data from those cores. Changing CO2 values affect the PH of the ocean, and this has various effects on the plankton fossils and carbonate chemistry. One effect is that the chemical speciation of boron in their shells will be affected. High CO2 dissolves in the ocean as carbonic acid, acidifying the water and causing there to be more B(OH)3 relative to B(OH)4. B(OH)3 is relatively more likely to form with the heavy isotope of Boron, Boron-11. Several other isotope systems can be used in similar ways. For example, carbon and oxygen isotopes (d13C, d18O) of foraminifera fossils have also been used to reconstruct atmospheric CO2. The processes determining these values are complex and involve ocean PH and partial pressures of dissolved gasses. The concentration of a chemical called alkenones in photosynthetic plankton is one of the more recently invented deep-sea core proxies for CO2 as well.
One of the non-isotope proxies for CO2 dating back tens of millions of years is the stomatal index record. Stomata are microscopic "mouths" on plant leaves that open and close to let in CO2 but keep H2O from escaping. In a higher CO2 environment, plants grow more stomata per unit area. Stomata can be counted in some plant fossils, and this can be used as a proxy.
Most of the proxies are too complex to be modelled backwards to get the atmospheric CO2 values that caused them, so they must be calibrated against more "absolute" records such as bubbles in ice cores.
One method is through ice cores from the worlds ice caps.
Each year, as small amounts of snow accumulate on ice caps such as on Antarctica and Greenland, bubbled of air gets trapped. As we drill through the ice, we can identify the air samples in those trapped bubbles, and measure directly the composition of the air in those bubbled. It tells us much more than the level of CO₂. For example, the ratio of isotopes of oxygen is a pretty good thermometer. We can also see pollen, volcanic ash, and other things.
Other methods are listed at the Wikipedia article on climate proxies. Most famous are ice cores and tree rings, but other methods are lake and ocean sediments, corals, and others. Those methods are somewhat independent so if they confirm each other, that is good. Errors on timing may get larger if one gets further back, and sometimes a new analysis leads too a change in timing estimates. But for the period where we have comparable records (for example, the period for which written historical records exist), results compare pretty well!
Huh, this is a very interesting question. According to a research paper:
The long-term carbon cycle is controlled by chemical weathering, volcanic and metamorphic degassing, and the burial of organic carbon (1, 2). Ancient atmospheric carbon dioxide levels are reflected in the isotopic content of organic carbon (3) and, less directly, strontium (4) in marine sedimentary rocks; the former because photosynthetic carbon isotope fractionation is sensitive to CO2 levels, and the latter because weathering and degassing are associated with extreme values of the abundance ratio 87Sr/86Sr. However, attempts to use these geochemical signals to estimate past CO2 levels (5–8) are hindered by the signals' additional relationships to various tectonic (9, 10) and biological (11) effects. Moreover, the strontium signal has proven especially difficult to parse (12–15).
So sometimes, carbon and strontium can be used to measure the CO2 levels. This is because CO2 levels are reflected in carbon and strontium. However, this is hard to identify, but in the paper above, people have studied what the CO2 levels were via strontium and carbon.
In addition, sometimes ice cores in the Arctic will be drilled, because they trap air bubbles from long ago. Scientists can then measure the amounts of various contents in this air, including CO2. They can represent atmospheric conditions up to 160,000 years ago.
If you'd like a more general overview of climate proxies, I'd recommend reading this website.