To answer the original question, yes its possible gravity cannot be unified with the other forces, although I'd be very hesitant to say it can't be quantized (we don't have evidence for it but a 100 years of success with quantum theory point to quantization along with a host of theoretical thought experiments). The question you're asking is the most fundamental one in physics and no one has the answer.
Yes, classically gravity is described in the language of geometry. However there is another perspective and in fact a consistent formulation of a QFT for a spin two particle will lead to Einsteins equations in the classical limit. So yes the geometrical interpretation came first but then came the more quantum interpretation.
I would also like to state that Gauge theories can similiarly be described be geometrically (at least classically I think...) using some more complicated ideas (fiber bundles and connections thereon). Just because something is classically geometrical doesn't mean it can't be quantized. (QED and QCD are quantum gauge theories)
Finally its likely that spacetime as we know it is an emergent property, it only makes sense at large scales and not at the Planck scale. Don't let classical formulation be an obstacle for a quantum one!