A necessary requirement for a traveling wave $u(x,t)=f(x-ct)$ to be a "solitary wave" or "soliton" is that the two limits $\lim_{s\rightarrow\pm\infty}f(s)=\alpha_\pm$ exist. This is the condition of shape invariance and localisation. The stability under collision may or may not be added as extra condition, but in much of the literature any shape-invariant localised wave is called a soliton. One further distinguishes homoclinic and heteroclinic solutions, depending on whether $\alpha_+$ is equal to $\alpha_-$ or not.
In response to Q2, the waves $u(t,x)=e^{i(x-t)}$ or $u(t,x)=\frac{1}{1-\frac12 e^{i(x-ct)}}$ are no solitons, because they are not localised.
Concerning Q1, without the "stability upon collision" condition, one way to identify solitonic solutions of a second order wave equation $f''(s)=F[f(s)]$ is to plot the flow lines in the f-g plane of the two coupled equations $f'(s)=g(s)$, $g'(s)=F[f(s)]$. Homoclinic or heteroclinic orbits then correspond to solitonic solutions.