Yes, we should agree on a semantic convention to prevent that we argue about words. When I say "know" or "fact", I'm referring to observations. When I say "assume" or "suppose", I'm referring to a hypothesis.

Of course you can also argue about how to interpret observations, but here I'm going with mainstream science.

The lack of oxygen in the early atmosphere is based both on some logical consideration (as long as there's no life, there's also no photosynthesis and thus no source for oxygen) and geological observation (molecular oxygen is only found in minerals younger than 2.5 billion years). In the Miller-Urey link I've posted above you'll find some information about the early atmosphere.

The estimate of several hundred million years for the evolution of life up to the first prokaryotes is also based on observation. It's the time difference between the oldest minerals on earth (4.4 billion years), and the oldest minerals in strata containing fosslized prokaryotes (3.5 billion years). The method to determine the mineral age was the Uranium/lead method, which has an inaccuracy below 1%.