not sure what you are talking about. we never did much with graphing Laplace transforms when i did differential equations. not sure exactly how such a graph would relate to the problem per say. maybe you could elaborate
4. Finally, Am I miss something, I understand that with Laplace transformation you can for example digitalize sin / cosine curve and any analogical curve. However if I draw Laplace tranformed function I cant see any digitialization in chart. What Laplace function represents actually?