Why "Nechceme ani tamto, ani tohle" is not accepted?
no, it switches the order
"ani tohle ani tamto nechceme" was not accepted. I assumed this could have been an acceptable word order in Czech.
Why "Nechceme" and not "Chceme" ?
Because the English sentence is negative, we do not want those. And in Czech we do not fear double negatives.
I'm guessing if I wanted to say, "We don't want this or that," it would just be "Nechceme tohle ani tamto" without the first ani and that having the first ani denotes "neither this nor that?"
"We do not want this or that." is also accepted in the reverse exercise for "Nechceme ani tohle, ani tamto.".