Both, Either, Neither

I have a few questions on https://www.englishgrammar.org/both-either-neither-2/ . I hope you don’t mind helping me.

10. ..................... him nor his parents were able to give a satisfactory explanation.

I think this question is grammatically wrong. Do you agree? It should be “he nor his parents was able...”

11. I have invited ................... of them.

either

both

neither

I choose “neither”, it says wrong and it should be “both”. I would say both answers are correct. What would you reckon?

12. I wrote to ................... of them but .................. replied.

both, neither

either, both

either, neither

I choose “both, neither”, it says wrong and it should be “either, neither”. I believe mine is still a valid answer. I do not quite accept their answer. If I wrote to either of them, that was meant only one of them. How could we say “neither replied” then?

By the way, do we say “ head or heads of men”? I’m referring to all the heads of several men. I’m not sure since every man has one head.