When it's winter in the Northern Hemisphere, it's summer in the Southern Hemisphere, and vice versa. It's just the case with the Earth than most of the land masses are on the Northern Hemisphere, but would not necessarily be the case with another planet.
Conflating two different effects. What you describe is due to the Earth's tilt. There's a smaller but globally significant effect due to moving closer/farther from the Sun in Earth's elliptical orbit.
Yes, but it's even more complicated. According to Wikipedia (https://en.wikipedia.org/wiki/Oxigen#Photosynthesis_and_resp... ), the green algae and cyanobacteria in the ocean produce between 70% and 45% of the free oxygen. (It's a lot of variation in the estimation.)
True, but just measuring carbon dioxide for the whole planet does show a distinct seasonal variance and that's for the reasons previously stated - there's a lot of land mass in the northern hemisphere and terrestrial plant life has a significant effect.
Possibly on exoplanets there might be similar asymmetries, or the asymmetry of an elliptical orbit. Seasonal variation in atmospheric gasses aren't necessarily a certain sign of life since there could be other non-biological causes, but they're a big clue.
