So my understanding is that to temper chocolate, it requires manipulating the heat to form the ideal crystal structure (i.e., beta V) that has the ideal properties for chocolate. These temperatures are different depending on the type of chocolate. Why would that be the case? I understand that milk and white chocolate contain different ratios of cocoa solids, cocoa butter, milk fats, milk proteins and sugar, but how do they affect the required temperatures for a successful temperature. Many articles explain that the tempering process allows ideal crystals to form and undesired crystals to "melt away". However my understanding of melting points is that the melting point of matter stays the same regardless of whatever it is mixed with. So if we heat dark chocolate back up to 31-32°C to "melt away" type I-IV crystals but keep the type V crystals,  how does having a different composition require milk or white to melt away the undesired types at different temperatures? 
When it comes to thermodynamics, you always have to consider the whole mixture, not its components separately. This is true of everything, not just chocolate. Mixtures behave in their own way - not entirely independent of their components, but the interactions within them are complex enough that you need some serious chemistry to start predicting melting or boiling points of simple mixtures (e.g. the boiling point of a mixture of known proportions of two liquids, as long as they behave close enough to an ideal liquid). For something as complex as chocolate? Forget it. Just go in and make empirical measurements. And why does nature behave so? Imagine the solid chocolate as a quilt. All the components have to fit exactly together to make the pattern. But there is no quilter to assemble the pieces; they float around, bump into each other and, if the conditions are right, the bumped sides stick together. Then other pieces start fitting into the already-combined mini-lumps, and they have to be the right pieces too - in a quilt, you cannot fit a hexagonal element into a space with 90° angles. This is why chocolate tempering is easier when "seeded" with tempered chocolate - you already have the first crystals onto which the new molecules have to latch in the right pattern, while without seeding, you have to rely on providing the conditions under which the spontaneously formed initial crystals are mostly of the form you want. In this model, milk chocolate and white chocolate will have different "quilt patterns", so they will need different conditions for the sticking-together (crystal growth) to happen. And temperature is one of the most important conditions in melting/solidifying, so it is no wonder that you need a different temperature for the two. 