Finding the system output by convolution
You can't compute the output by convolving the input with the system's impulse response if the initial conditions are non-zero. The reason is simple: there is no information about the initial conditions in the impulse response or, equivalently, in the system's transfer function.

What you can do is use the following properties of the unilateral Laplace transform:

$$\mathcal{L}\{y'(t)\}=sY(s)-y(0^+)\tag{1}$$

$$\mathcal{L}\{y''(t)\}=s^2Y(s)-sy(0^+)-y'(0^+)\tag{2}$$

If you use $(1)$ and $(2)$ when you transform the differential equation, you'll get an equation for the Laplace transform of $y(t)$ which takes into account the given initial conditions. The inverse Laplace transform of that expression gives you the desired system output for non-zero initial conditions.

As for stability, causal systems with a rational transfer function are stable if all poles (i.e., the zeros of the transfer function's denominator) are inside the left half-plane.