One of the most frustrating things about writing about the Great Wall on AskHistorians is that it is an incredibly prominent motif in popular culture, but there has not been a comprehensive academic work published on the subject of its physical history since Arthur Waldron's in 1990. While there are two more recent academically-authored works with 'Great Wall' in the title, one by Julia Lovell from 2006 and one by Carlos Rojas from 2010, these are cultural histories and not specific discussions of wall-building policy (for which Lovell draws heavily on Waldron anyway). So the answer below mostly reflects the academic understanding ca. 1990, which to be fair we could consider to still be the understanding today in the absence of an easily locatable update, supplemented with some material from the late 00s.

To give a very general overview, we need to understand that the Great Wall as it exists in the popular imagination is entirely a product of the Ming empire, which engaged in wall-building primarily over the course of the late 15th and 16th centuries. While there are considerable archaeological remains of prior walls, these 'walls' are in fact relatively spartan earthworks consisting principally of a ditch and embankment, without substantial masonry reinforcement like the walls that still stand today. The extent to which these Ming walls were necessarily deliberately built on top of existing structures is... controversial, but it is not unreasonable to suggest that the ideal sites for fortification remained pretty consistent over time. Nevertheless, the notion of a Ming 'restoration' of the Great Wall is problematic – not all of the Ming walls followed the path of prior structures, and even those that did most likely built on top of structures that were so eroded as to be effectively nonexistent anyway. This wall system was not, however, conceived of as a singular system to be built contiguously. Rather, wall-building was done at the regional level at different times along the northern frontier, and was intended to supplement defences whose principal element was semi-mobile armies, with the walls serving to slow raiders and invaders or to funnel them into more advantageous terrain for the defenders. That is why there seem to be so many 'gaps' in the Ming wall system – those are the product of both its quite disorganised construction history and its design philosophy.

We also need to consider that wall-building tended to serve the specific purposes of specific states, and states that did not see themselves as needing walls to defend against frontier threats (either in the absence of such threats or out of the presumed existence of more cost-effective solutions) would simply abandon these walls. There was no substantial wall-building in China between the 2nd century BCE and 6th century CE, and again between the 7th and 15th centuries, for the simple reason that successive states saw no need for them. What the map describes as 'construction' is probably a little deceptive: if three successive periods are shown to have carried out 'construction' in the same area, that probably actually just indicates continuous occupation and consequent maintenance.

So then to answer your specific questions in turn:

How can you tell what is "main" and what is not? Is it just the longest unbroken portion of wall? Is it the part that was considered the most important from a defensive point of view?

Simply put, this is the rough length of the Ming wall from its western end at the Jiayu Pass in Gansu to the Shanhai Pass in Hebei, excluding the Liaodong Wall (largely a more austere structure that was mostly earthworks and palisades) that extended into Manchuria. This wall structure was the most elaborately built and maintained, being mostly masonry walls rather than simple earthworks. How 2400 was calculated is not exactly clear, as given the pitfalls of the coastline paradox you can give some wildly varying numbers depending on how precise your gradations are. Still, if you were to drive from Jiayuguan to Shanhaiguan on modern roads you would end up covering about 2300km per Google Maps, so I do want to just affirm that the 2400km number doesn't come from nowhere, and would be around what you'd expect from a low-resolution estimate.

Why do some portions of wall look doubled up? Does that just mean that one dynasty rebuilt a portion of wall from a previous dynasty that had fallen into disrepair? Are there literally two walls next to each other?

So this is where the limitations of the scholarship come up, because it could be that the map is based on more up-to-date information than is easily accessible. However, based on Waldron and others, there are some reasonable explanations, for different spots. I'll broadly go chronologically here.

The stretch between Guyuan and Yulin appears to have both Warring States and Qin walls, but this is effectively just to indicate that the structure was used by both the Kingdom of Qin and its immediate successor state, the Qin Empire.

The wall on the north bank of the Yellow River probably was newly built by the Qin, assuming that we can parse the language in the historical accounts as referring to Qin Shi Huang ordering the construction of 'a wall around the cities' and not 'walled cities'. Wall-building further eastward under the Qin seems dubious and instead the map likely just reflects Qin occupation of walls from the Warring states period. The same goes for the Han, where the historical record suggests the existence of walls on the Han frontiers, almost certainly held over from the Warring States, but not the construction of entirely new structures outright.

The Han did build new walls in Gansu, but the Ming ones don't 'double up' with these for the most part. Even where they do, this is easily explained by the general topography of the region – and thus the most defensible positions for the territory that most needed defending – being effectively the same.

The Northern Qi walls at Xuanfu-Datong are perhaps better understood as part of a large local fortification network rather than as a frontier wall, and the continued occupation by later states is pretty comprehensible given the location's defensibility. As can be seen, the Ming walls only cover the northern part of this area, and not the southern, because of their function as northward-facing frontier defences, tracing the line of older fortifications already built on good defensible ground.

Why are some portions of wall from the same dynasty doubled up? Was this for extra defense? Did they lose/gain territory and build a new portion of wall to defend the new border?

This gets back to my general overview from the start. If I'm not mistaken there are two main areas of 'doubling up' visible on the map: the presence of Qin walls both south and north of the Yellow River, and a few stretches of Ming wall that exist in front of what seem to be the 'main' structure. Both of your explanations are correct, but each for a different situation. The Qin Empire definitely did expand over the Yellow River to control territory it had not held as a kingdom, and so built new walls to encompass these conquests. The Ming, on the other hand, were deliberately aiming at building layered defences in places.

What on earth is that chunk of wall doing way up in Mongolia/Russia? I know that Mongolia and China were at one point part of the same empire but that piece seems so far removed from the rest.

The answer is that the map is sneakily claiming the Khitan Liao and Jurchen Jin states as 'Chinese' ones. In reality these states were founded by peoples from the region sometimes known as the Northeast and sometimes as Manchuria, and they were primarily based in that region, which historically lay outside the effective control of most Chinese states. Only the latter would ever exercise rule over a substantial amount of China proper. The earthwork systems built and maintained by the Khitans and Jurchens (the latter of which were quite sophisticated) follow the line of their frontier with the Eurasian steppe, rather than being from either of the two states (Yuan and Qing) which controlled the entire region (as these states also controlled Manchuria anyway).

The orange portion of wall seems really long for how little space it takes up on the timeline at the bottom. How'd they get so much done in 15 years?

There are in effect two sides to this. The first is the simplicity of the Qin-era walls, which were simply earthworks that could be completed in a matter of weeks. It was not that hard to build one. The second is my suggestion – though I would like to think a reasonably-supported one – that the map displays continued occupation of a wall system as 'construction' when in reality it is more like 'maintenance and limited upgrading'.

Why are some of the purple sections more like a dotted line rather than an unbroken line?

Again, there are I think multiple, potentially complementary explanations. The first is that the Ming wall was never intended as a contiguous structure. The second is that the Ming system, being substantial masonry-faced works, was much harder to build, and thus filling the gaps was not always considered cost-effective. The third I think is the most important, which is that our information about the older walls is pretty patchy, whereas the Ming wall has left very easily located traces. Paradoxically, this means that the map shows a lot of gaps where we have high certainty and none where we don't. I would suggest that part of the reason why the older walls are displayed as contiguous is that if there are gaps, we simply don't know where they are, and so the map is tracing their general route rather than specifically marking where we actually know walls to have existed.