Does it mean both:
"She doesn't want to allow it"
"She will not allow it"
Yes, it means both. Context is important here.
And would "Hun vil ikke lade det also be correct? If not, is there a difference?
I'm not 100 percent certain but think that gives you "She will not leave it"
"Hun vil ikke lade det" would not really make much sense but "hun vil ikke lade det SKE" could be used just fine instead
No, that would mean "She will not let it" which doesn't make sense.
when I put down Hun vil ikke lade it was marked wrong.