Does static longitudinal stability require download on the tail?
No, static longitudinal stability does not necessarily imply a download on the tail. 



Static longitudinal stability requires that the Centre of Gravity is in front of the Centre of Lift, indicated as n.p.$_{fixed}$ in the drawing. Only then will an increase in Angle of Attack d$\alpha$ result in an opposing pitching moment: if d$\alpha$ > 0 then $dC_N$ > 0, resulting in a higher lift force. The neutral point must be behind the CoG in order to transform the higher lift into a nose down moment. Note that the drawing shows a longitudinally stable situation with a positive lift on the horizontal tail.