Was the Han Dynasty of China related in any way the the Han State which existed during the Warring States Period? If so, how were they capable of emerging victorious in the civil wars that occurred after the fall of the Qin Dynasty?

To make a short story as short as it really is, the answer is no, as the characters are distinct both in writing and in tonality. The 'Han' in the 'State of Han' is rendered with the character 韓, and pronounced Gar in Old Chinese, Han^2 in modern Mandarin and Hon^4 in Cantonese. By contrast the 'Han' in 'Han Dynasty' is rendered 漢, and pronounced Nar in Old Chinese, Han^4 in Mandarin and Hon^3 in Cantonese. As to why Han, that is 漢, triumphed, I'm afraid I must defer to the wisdom of flairs familiar in this area such as /u/cthulhushrugged and /u/Jasfss.