Is there a particular reason "nas" comes before "nie"? Would it also be correct to write "Ten mężczyna nie widzi nas"?
Yes. Puts a bit different accent (to "us"), but it's perfectly correct and accepted here.
ten mężczyzna - subject, nominative
nie widzi- negated verb, widziec needs accusative, so negated needs genitive
nas - here genitive, "my" has genitive=accusative
A bit less natural, stress on 'us', but definitely acceptable. And it works in the ENG-PL exercise.
Where does the object pronoun go located? Lubie to for example the object pronoun goes after the verb but in here it goes before.
If I say "Ten mężczyzna nie widzi nas" is still grammatically correct? Or only this happens in a negative form?
Is there a rule to know for the rest of the object pronouns in Polish?