It is well known that the blow-up of $\mathbb P^2$ in one or two points does not accept a Kahler-Einstein metric. Kahler-Einstein metrics are particular cases of constant scalar curvature Kahler metrics (cscK metrics). I was wondering if:

1. Is it known if the blow-up of $\mathbb P^2$ in one or two points accepts a cscK metric?
2. Are there examples of complex projective manifolds (or varieties with klt singularities) that do not accept any cscK metric for all polarisations? I would expect these varieties, if they exist, to be Fano.

References are welcome.