Are "nama hari-hari" and "nama-nama hari-hari" also acceptable?
I would say all are correct. You could say in English "Say the name of the days". Which I think is a case of name being implied plural. Quite rare for English but common in Indonesian.
Why sebutkan and not sebut?
I think sebut is also acceptable. -Kan, which is a benefactive suffix, emphasizes that the names are said for someone.
(KBBI defines menyebutkan as menyebut (untuk orang lain), i.e. to say for someone.)