In the total beginning of the Earth there was CO2, N2 and H2O. So there was O2 but always in bond with Carbon and hydrogen, but what is the reason for that? Is O2 always reacting with free carbon atoms? The temperature of the Earth was is those beginning days around 1200degree Celsius. Is that also a factor of those bindings?
The 1200 C temperature is a big clue, given the auto-ignition temperatures of most "stuff". https://en.wikipedia.org/wiki/Autoignition_temperature
Any "free" oxygen would immediately "burn" with carbon and just about anything else, yes.
A complete answer gets long and above my pay-grade, but when the earth and solar system were young, it's unlikely that the Earth had much atmosphere at all. The young sun was hot enough to burn off most gases and ices in the inner solar-system and when the earth coalesced it was largely metals and silicates. The oceans and atmosphere came later by comet and asteroid belt impacts, and, as Jeff Y points out, after that it's just heat and chemistry. Oxygen or O2 binds so easily with other molecules, like Methane or Hydrogen or Iron that it goes away pretty quickly until those are used up.
Like O2, N2 wasn't nearly as present in the young atmosphere as it is today either. More on that here: http://sciexplorer.blogspot.com/2012/01/earths-atmosphere-part-4-evolution-of.html The four most common early ices/gases on earth were CH4, CO2, H2O and NH3. Over time, the NH3 and CH4 get used up by bacteria or chemical processes, lightning, volcanoes, UV light from the sun, etc. The H20 and CO2 are much more stable, though photosynthesis did slowly pull most of the CO2 from the atmosphere, returning O2.
Oxygen is an extremely eager element to bind to other elements.
There was, in fact, a lot of oxygen. But most of it was stuck to carbon etc., and later tied up in various ores and minerals - in general, always stuck to something else, until biological processes started "liberating" it (and, interestingly, causing a bit of an ecological catastrophe as free oxygen was toxic to the very microorganisms that produced it. It was not a catastrophe when considered from our point of view, obviously.)
TLDR, think of oxygen atoms like stray pieces of duct tape floating around.