What is the etymology of the word nǚ (woman) and is it related to Proto-Uralic *niŋä (woman)?
It is at best a coincidence. nǚ and niŋä are not even that close. There is only one common phoneme between the two words. If you count this as close, there will be thousands of words that are as close.
Also, today's pronunciations can't be used to compare, because pronunciations change over time. You can read the basics at https://en.wikipedia.org/wiki/Old_Chinese and https://en.wikipedia.org/wiki/Sino-Tibetan_languages. Basically most scholars agree that Chinese is related with languages like Tibetan and Burmese. There are also a very few scholars who support hypotheses like the &quot;Sino-Caucasian&quot; hypothesis, but these are viewed as doubtful and not widely accepted.