Without the right load caps the frequency won't be in-spec for that crystal (they are calibrated for a given load capacitance).
Depends on what you meant by "in-spec".
Load caps make the oscillation frequency much less sensitive to stray circuit capacitances I believe - not necessarily important.
It doesn't make it less sensitive. It makes the frequency closer to spec.
A particular oscillator design may assume a certain range of load capacitance too - without it there might be no guarantee of oscillator start-up?
Yes, but generally only for low frequency oscillators.