Does it make sense to do an emergency landing without nose gear in order to reduce stopping distance?
(Source)

That was the case on July 23, 1983 (but not quite).

It was luck the manual gravity-extension didn't lock the nose gear in position. That has been attributed to the shortening of the landing roll/slide.


  Two factors helped avert a potential disaster: the failure of the front landing gear to lock into position during the gravity drop, and the presence of a guardrail that had been installed along the centre of the decommissioned runway to facilitate its use as a racetrack.


However, no commercial aircraft is certified for it, because it can do more harm than good. And depending on the aircraft type, separate controls for each landing gear in manual-extension mode may not exist.

A hard landing without the nose-landing gear may rupture the fuselage, center fuel tank, or hydraulic lines. And with the added sparks from the sliding, that's a recipe for disaster.

Instead, a system such as the engineered materials arrestor system (EMAS) may help:

(Source) EMAS.