The Hebrew Bible is among the early texts mentioning Susa (under the Hebrew name of Shushan which is the same world as a lily) but there exist somewhat older, Sumarian documents that mention Susa, e.g. Enmerkar and the Lord of Aratta where Susa is quoted as a place obedient to Inanna, a goddess of love. This text is from the 21st century BC but similar Sumerian texts going back to the 25st century BC may exist. The Sumerian civilization probably existed 4,000 BC and could have used the "same" words. They began to write around 3,000 BC.
Susa comes from the name of the local deity, Inshushinak. Conveniently, this word "works" in English. God Inshushinak was "in Shusha". Most sources logically say that the relationship is the other way around: Inshushunak "means" the lord of Susa.
The Sumerian language – the most ancient known written language – was a language isolate which makes conclusions about the relationship of their words to words in other languages particularly problematic. (Can the beginning "in" of the deity's name be identified with "in" in English, for example?) It is hard to claim that its words are related to words such as "six" if linguists don't even agree to which modern languages Sumerian was closer.
It's easy to get carried away and propose relationships most of which are almost certainly coincidental. For example, šuša could be claimed to be related to the Czech word "šuškat" (whispher) or "šiška" (conifer cone) or "sušák" (dryer) or "sušit" (dry) or "sušenka" (a dry cookie) or even "šůsovat" (slang for fast direct Alpine skiing) but is there any reason to think that the similarity is more than a coincidence? Most of such short combinations of sounds are being used in every language.
The Sumerian word has affected lots of names in the Elamite culture. The Elamite language was another language isolate, however.