If I understand your question correctly, it is about measuring the impulse response or frequency response of an actual system. First of all, usually the transfer function of a system is defined as output divided by input, instead of the other way around as in your question, but that's of course just a convention.
One of your questions appears to be "can I use any input signal to measure the system's frequency response (or impulse response)?" The answer is clearly no. As an extreme example, take a sinusoidal input signal $x(t)=\sin(\omega_0t)$. An LTI system with frequency response $H(\omega)=|H(\omega)|e^{j\phi(\omega)}$ will have a response $y(t)=|H(\omega_0)|\sin(\omega_0t+\phi(\omega_0))$, where $\phi(\omega)$ is the system's phase response. Consequently, with a sinusoidal input signal you can just measure $H(\omega)$ at a single frequency (namely the frequency of the input signal). That's why an impulse is a good input signal for measuring the properties of a system: it contains all frequencies, not just a few.
One common practical method to measure a system's impulse response (or frequency response) is the sine sweep method, where the frequency of the input signal is swept over the whole frequency range of interest. This method and a few others are compared in this paper.