Killing spinor equations are equations that result from supersymmetric transformations. One example of those is for example

$$\nabla_{\mu}\epsilon+\frac{i}{2}A_{\mu}\gamma_*\epsilon+\frac{i}{4}Im\mathcal{N}_{AB}\gamma_{\mu\nu}G^{\mu\nu A}(Im L^{B}-i\gamma_* Re L^{B})\gamma_{\mu}\epsilon=0$$ This is the usual one present in $N=2, d=4$ Supergravity theories.

As suggested by some books and papers on the web, this is the vanishing of the gravitini supersymmetry.

My related questions are:

1- Why do we have to solve these equations known as killing spinor equations? In other words, how are their solutions important or beneficial in any way?

2- I understand that there in supergravity there is a graviton, 2 gravitini and a 'so-called' graviphoton. So if fermions here should vanish in $N=2$ supergravity theories (those coupled to vector multiplets), why don't we see vanishing of the graviton supersymmetry or gravi-photon supersymmetry but instead we see vanishing of gaugino supersymmetry? Where am I confused about here?