Does "improper" imply that a system cannot be stable and causal?
An improper system cannot be causal and stable. If the order of the numerator is greater than the order of the denominator, you'll always have at least one pole at infinity. Consequently, not all poles are in the left half-plane (or inside the unit circle in the case of discrete-time systems).

The system in your example is clearly unstable:

$$H(s)=\frac{s^2}{s+1}=s-1+\frac{1}{s+1}\tag{1}$$

Part of it is an ideal differentiator ($s$), which is unstable.

Also take a look at this related answer.