Impulse response of a linear system
Can we identify the order of the system (the order of the differential equation)

Yes , we can identify the order !
One method could be to find  Laplace transform of $$h(t)$$
And  then  degree of denominator of polynomial $$H(s)$$ would be order of differential equation .

and the differential equation itself?

Yes ! Use  $$H(s)=Y(s)/X(s)$$ then separate Y(s) and X(s) and then take inverse Laplace transform assuming Zero initial condition and you'll get desired differential equation.