Re: Transfer Function.
The transfer function is really the ratio of the Laplace transfor of the output to the Laplace transform of the input. Therefore, it is not the gain that you would measure in the time domain.
The Laplace transform becomes a tool that allows you to more easily solve differential equations. You transform all quantities involved, do the calculations, which thanks to the Laplace transform only involve simple arithmetic, and then transform the result back to the time domain. The transformations are relatively easy to do, thanks to tables of Laplace transforms.
The Laplace transform actually transforms the time domain into the complex domain, more than just frequency. That allows you to also use it for transient responses, that is, non-periodic signals.
You do not have to worry too much about its physical meaning, consider it just a mathematical tool.
Perhaps an analogy would be the old slide rule: it allows(ed) you to do multiplications and divisions, simply by addition/ subtraction. Of course, it has to use logarithms to achieve that, but in the end, you do not care about the logarithms the slide rule is based on, you just care(d) about the ease of doing a relatively complicated operation, such as division, by simply subtracting two quantities. In a way, you "transformed" the inputs to their logs, added/ subtracted these "transforms" and finally read the result, "converting" it back to a humanly-understandable number. The "transformations" were transparent, of course, but they existed, they were embedded in the non-linear divisions of the slide rule.
The above is not meant to belittle Laplace's contribution to science. I tip my hat to the great man.