1. For each piece, you can regard the Heaviside step function as a switch that turns that piece on or off. I think you need to have something more like the following:
2. Your Y(s) is correct. I'm not sure what your last line is doing, though. Why not just take the inverse transform of what you have? That's the beauty of the LT approach: it takes care of initial conditions, piece-wise defined functions, and inhomogeneous terms in a fairly straight-forward manner, when it works at all.