The relationship between completeness and strength of reductions
If I understand correctly your questions are answered in 

Agrawal-Allender-Impagliazzo-Pitassi-Rudich-2001: Reducing the Complexity of Reductions


Gap Theorem - Any set that is NP-complete under $AC^0$ reductions is NP-complete under $NC^0$ reductions. 
Stop Gap Theorem - There is a set that is NP-complete under $AC^0[\!\!\mod 2]$
reductions, but not under $AC^0$ reductions.