Why does lithium from the nitride whereas the other alkali metal forms the azide?
The formation of $\ce{Li3N}$ can be explained thermodynamically. The enthalpy of formation of the $\ce{N^3-}$ ion is extremely endothermic (it involves three successive additions of electrons) and with the larger counterions $\ce{Na+}$ or $\ce{K+}$ the lattice energy (which is proportional to $\frac{q_+ q_-}{r_+ + r_-}$) is not sufficiently large to compensate for the large $\Delta_\text{f}H$ of $\ce{N^3-}$. The same argument explains why magnesium also forms a nitride.

According to my very brief research on the Internet, lithium azide does exist but is unstable. It can be prepared via reaction of $\ce{NaN3}$ and $\ce{Li2SO4}$: see Acta Chem. Scand. 1957, 11, 581 for more details.