well, sure. otherwise you get [itex]Y(s) = G(s)U(s)[/itex] + some other stuff.
sure, if you want to represent the (transformed) output as only the transfer function times a single input, then you have to assume that the system is completely "relaxed" at time 0. but you can
model a system in terms of only the input and output (and their initial conditions) with the Laplace Transforms of the input and output. but it won't be a simple transfer function.
the main difference between this and the state-space model is that the state-space model is trying to model what is going on inside
the box. it is more general than the simple input-output description of the system.