What was the atmosphere of the Earth like after the sun went through the T- Tauri stage?

This is hard to estimate because it depends on what you think the formation process was. An older view was that it might have been rich in methane, ammonia, water and other gaseous compounds we know about from studies of interstellar clouds. These compounds are also found in comets. Comets, or similar bodies, may even have been the primary delivery vehicle for these volatile compounds. The solid earth may either have formed cold, but as it grew larger the action of trapped radioactive nuclides decaying heated the interior up. Large 500 - 1000 km bodies impacting the surface probably left the surface in a near-molten state with a surface temperature of thousands of degrees. These impacts probably continued well beyond the T-Tauri turn-on stage of the sun which happened about 10 million years after the initial cloud collapse. The estimates I know about suggest that the T-Tauri wind was powerful enough to strip away any primordial atmospheres that existed leaving the earth a naked ball of rock. But cometary nuclei rich in volatiles were probably un affected beyond the orbit of jupiter and continued to rain down upon the inner solar system for millions of years after the short-lived T- Tauri phase ended, reinstating an organic molecule-rick, reducing, atmosphere. Outgassing from the planet also contributed trapped interior gases rich in CO2 and sulfer which were locked up in the compounds out of which the earth was formed. Then about 60 million years later, the earth was again impacted by a large body which formed the moon, and probably stripped most of the atmosphere away again. Perhaps a third epoch of comet impacts refilled much of the atmosphere...perhaps even the oceans, and then a few million years later life arose from this stew somehow.