I have been recently exposed to the idea that there can exist solutions to systems with no analytical (symbolic) solutions. However, I could not find any source to the question of whether a second-order non-linear ODE of the form: $$\ddot{x}(t) + \alpha \dot{x}(t) + g(x(t)) = 0$$ can have a non-differentiable solution, where g(x) is a non-linear function which is everywhere continuous and differentiable as well as integrable and $\alpha \in \mathbb{R}_ {>0} $.
Also are there any methods to prove that all solutions must be everywhere differentiable within the domain $t>0$ ?
By definition a solution to this equation is a function whose second derivative satisfies the equation. Therefore, the solution has to be twice differentiable. There are extended concepts of solutions which can fail to be differentiable, when the right-hand side contains objects such as impulses.
I don't see how this could happen in your equation, as g is continuous. A solution may fail to exist if g doesn't respect Lipschitz conditions (roughly, if the derivative of g is too large with respect to g), but if a solution exists, it is continuous.