Question regarding transfer functions and prerequsities for finding the real impulse response The transfer function of a system is given by:

$$\large H(s) = \huge \frac{V_{out}(s)}{V_{in}(s)}$$

In digital domain the principle is of course the same, just replace laplace transform with z-transform and voltages with amplitudes. The transfer function, when converted to time domain, solves the impulse response of the system when conditions are met.

My question is about the prerequisites for finding the real impulse response using this method (other than LTI properties). Obviously, if a signal with a shorter length than the impulse response is used for the analysis, the real impulse response can't be solved for. However, is it enough to have a signal the length of the impulse response, regardless of what the input is? (Using dirac delta as input will of course solve the impulse, but what about other cases such as $sin(t)$?)

I can at least answer the question when it comes to fourier transform and related frequency response $H(w)$. In that scenario the signal indeed needs to be zero padded properly or we will have circular convolution. I am assuming the case is then the same for laplace transform. Thus, is it correct to say that for the $H(s)$ to yield the correct transfer function, the time signal representing the input needs to be zero padded by the length of the impulse (-1)?
If I understand your question correctly, it is about measuring the impulse response or frequency response of an actual system.

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.