Indeed, an integer is divisible by 3 if and only if the sum of its digits is divisible by 3.
For proof, take the polynomial representation of an integer n = a_0 * 10^k + a_1 * 10^{k-1} + … + a_k * 1. Note that 10 mod 3 = 1, which means that 10^i mod 3 = (10 mod 3)^i = 1. This makes all powers of 10 = 1 and you’re left with n = a_0 + a_1 + … + a_k. Thus, n is divisible by 3 iff a_0 + a_1 + … + a_k is. Also note that iff answers your question then; all multiples of 3 have to, by definition, have digits whose sum is a multiple of 3
Like you said, there’s a metric ton of steps involved in the overall process, and our understanding of each of these ‘levels’ of organization varies quite a bit. Closest to my personal expertise is the sub-cellular and cellular level, for which I would refer you books or papers in developmental biology, developmental genetics, and epigenetics. I can give you a couple bloopers to get you excited though:
When it comes to deciding where different organs will form (again, from a sub-cellular level), it essentially starts from a concentration gradient of certain proteins/chemicals inside the zygotic cell. This gradient then determines the level of activation of other proteins, each level then leading to different biochemical cascades which, once more cells have formed from the zygote, determine which organ systems will come from them. Messing around with this concentration gradient can, thus, dramatically affect an organism’s development, leading to, for example, a body with just ‘left’ organs on both halves, just ‘top’ organs on both halves, missing an entire organ system (like circulatory system and heart) and so on.
A more or less similar process occurs to determine the shape of organs. As a simple example, when some animals with regenerative capabilities (like axolotls) lose a limb, they are able to regrow the limb to the exact same length as before. Turns out, each cell on the periphery of their limb has a certain concentration of receptor proteins on its surface, which acts as a molecular ‘signature’ of that cell’s position in the limb. These signatures provide information on how far to grow the limb for regeneration, and some chemicals, lile retinol, can even override these signatures and fool the organism’s body into regrowing the limb from scratch on top of the place of regeneration.
I hope these examples give you an introduction to the mechanisms involved. There’s obviously a lot more involved, so I would again highly recommend textbooks and research papers if you’re interested.