Laplace transform of product of signal and impulse train
$\delta(t)$ is indeed the Dirac delta impulse, which is not defined by its values (because it is no ordinary function), but by its properties under an integral.

The first expression in your question is a standard model for sampling a continuous-time signal, and since it's a mathematical model it does not represent actual physical sampling. You can replace the Dirac delta impulse by other impulse-like functions. If you use a rectangular impulse you get a zero-order hold. However, if you're not interested in any effects of non-ideal sampling, multiplication with a Dirac comb is a convenient and useful model for sampling.