When and how did Russia manage to claim such a large amount of territory and at what point were these claims considered legitimate by other powers?
Hi! I don't recall seeing much/any discussion on your last point, so hopefully someone can pitch in for that, but there have been several threads on Russian territorial expansion (as well as related topics like relations with indigenous peoples, the fur trade, governance) that you may be interested in:
What's the driving force behind Russia's expansion? - a recent (still open) post featuring a concise summary by /u/facepoundr
How did pre-soviet Russia get so big? - a Q&A-like thread featuring /u/archaeontologist
How was Russia able to form into such a large country? - a discussion thread featuring /u/ayyum and /u/facepoundr