Calculating steady state output using Laplace transform
Yes, for these Laplace transform problems, "constant input" translates to "unit step".

As a further check, you can reason through it -- while convolution is usually not something to try to do in your head, the convolution of a pulse with a constant is just the height of the pulse times the constant, integrated over the duration of the pulse.  So, 3 * 3 * 5 = 45, which is not only the answer you got, but which follows the arithmetic in your final few lines fairly closely.

(BTW: If I were a grad student grading your work you would have gotten full marks.)