How to find the steady-state response from the impulse response
Convolution simply gives you the output of a system given an input. In the case of a periodic input, you'll end up with a periodic output. This implies that the steady state response to this input will be a periodic "steady state" response. I put steady state in quotes because it's not really steady in terms of DC analysis. It is steady though in terms of the frequency domain.

To answer your main question succinctly: No (but almost), the steady state response means the output after the initial transient has settled out.

Taking some quotes from wikipedia may make it more clear:
"steady state is an equilibrium condition of a circuit or network that occurs as the effects of transients are no longer important"
"A transient event is a short-lived burst of energy in a system caused by a sudden change of state."

The burst of energy referred to here would be the u(t) that gives the change in state. The periodic function is actually part of the steady state nature of the system and it will affect the transient response and the periodic steady state of the system, but it itself isn't the transient event. It's periodic and so forever in the past and forever in the future will continue to do it's thing which in the frequency domain is definitely steady.

Let me know if that's still confusing.
There's some nice pictures on this page showing it better than I ever can in words:
https://ccrma.stanford.edu/~jos/fp/Transient_Response_Steady_State.html