"There are not many of these women."
Translation:Těchto žen není mnoho.
Why is the sentence contructed with this word order? And why is the singular used?
It is one of several possible word orders. Hard to say more, really, you have to ask concretely, if you have an actual problem.
Singular, because mnoho works like a grammatically singular word, although it is an adverb. (You can say "to" mnoho.)
well in many other sentences you often put the verb first, would this work in this case as well, and if it worked how would you construct the sentence then?