A very fair series of questions. I'll do my best to answer, but please feel free to follow up with additional questions if you have any, or if I've been unclear!

How were they only outnumbered 2-1 despite the population difference between them and their opponents?

To be clear, there were roughly 130,000-150,000 Egyptian troops arrayed against Israel in the Sinai. Syria had set up around 50,000-60,000 troops, and Jordan around 55,000 too. Israel, by contrast, had around 250,000-275,000 troops. You can see, in short, that the forces were not actually unevenly matched in terms of size, and Israel may even have had an advantage by most counts (historians vary). One of the main reasons for this is the Israeli draft: Israel was mobilizing over 10% of its population to fight, which is a huge order. Israel was able to do this because of its draft, which ensured that its very large reserve forces (over 2/3 of its forces were reserves) were still somewhat experienced and had been trained. There were also, as a result of the draft, plans in place for rapid mobilization of those reserves, and the reserves themselves were aware of those plans. However, the cost of mobilizing 10% of the Israeli population or so (and a larger portion of the Jewish population, since Israeli Arabs are not drafted and certainly very rarely if ever served in the IDF) meant that Israel needed a decisive strike and a swift end to the war to avoid a protracted conflict. This was also a big reason for its decision to launch a preemptive strike. Feeling encircled, rightly or wrongly in terms of how we view it from hindsight and in general, Israel had mobilized its reserves fully in response to the Egyptian mobilization (and mobilizations by Syria and Jordan). However, the Arab states could maintain those mobilizations for far longer. Israel could send its troops home and have them stand down, but that might leave it at a significant defensive deficit if Egypt had attacked after that demobilization. Alternatively, Israel could maintain mobilization but not strike first, leaving its economy to struggle more and more. This has been mentioned in numerous analyses and historical recountings. General Matti Peled, in one spirited argument over what to do while the forces were mobilized and waiting, said that the enemy was "digging in and getting stronger while our economy weakens and all for a purpose which no one has yet explained". Maj. Gen. Meir Amit said as much to US SecDef McNamara, pointing out the economic inviability of maintaining mobilization. So while Israel could field comparable numbers to Egypt/Syria, it could do so only at great damage to itself, while Egypt/Syria had no need/motivation to do the same themselves.

Another factor was that Egypt still had troops tied up in Yemen. While it had shifted some to the Sinai in preparation for a possible war with Israel (another big factor in the Israeli mind that a preemptive strike was indeed preemptive, since the Yemen troops were viewed as battle-hardened and more effective), many remained there. Egypt's President Nasser had himself pointed to the Yemeni troops as evidence he had no intention of attacking Israel, when speaking to the public. In 1965, he told PLO delegates that it wasn't conceivable that he would attack Israel "while there are 50,000 Egyptian troops in Yemen". But these troops had, as mentioned, begun to shift to the Sinai, albeit haphazardly and without significant resources.

The third factor, common to all the Arab states, was simply being cash-strapped. This, coupled with the training and planning points you've seen mentioned, worked in tandem to weaken the Arab armies. Egypt's GDP per capita was a fraction of Israel's, and Egypt's economy was seriously weak. Thus, the weakness caused by Israel's mobilization can be viewed in the context of Arab states' general economic weakness even prior to war. This was the result of factors like corruption, foreign politics, and the like, and could be a post of its own. At any rate, this corruption also fed into how the military forces were trained and set up. Israel's force was largely professional, drafted though it was, and set up in a more viable and merit-based structure. The heads of the Israeli military had combat experience and leadership experience. By contrast, Egypt's and Syria's were populated by politically-motivated appointments, meant to ensure loyalty to the authoritarian government (and prevent potential coups, or rather, sometimes were divided by jockeying to set up for a coup). Thus Egypt's military upper echelon was dominated by bureaucrats with little to no military experience. Maj. Gen. Gavish, the head of the IDF in the Sinai in 1967, put it thus: "[Amer, the Egyptian Chief of Staff leading the Egyptian military] created five new layers of command and with people who'd never fought. We'd be halfway to Suez before they'd get an order approved." Israel's leaders, as mentioned, were experienced: Gavish had fought in the Palmach (elite Israeli forces) during the 1948 War, for example. Israel Tal, a commander of an armored division in 1967 in the Sinai, had served in the Jewish Brigade under British command in WWII, an officer in the 1948 war, and a brigade commander in the Suez crisis in 1956. Ariel Sharon, who got the lion's share of credit for success in the Suez in 1967, had already experienced military success and controversy alike, serving in the IDF in the 1948 war, leading special forces units in offensive guerrilla war after that, leading a paratroopers unit in Suez, and leading an armored division of his own in the Sinai in 1967. The disparity could not have been clearer, as Israeli command was relatively clear and experienced and Egyptian command bureaucratic and political. Syrian command was no different. Hafez al-Assad (father of the current leader of Syria) was cognizant of potential coups, and often left military forces underfunded or under-armed for fear they'd turn on him if they were not amongst his most loyal. Thus his officers were also frequently loyalists, and complicated command made it hard for any to contemplate turning on him. Jordan had the most professional military, as in 1948 as well. However, Jordan placed its military forces under Egyptian command for the war due to heavy Egyptian pressure to sign a defensive treaty (one which also played a huge part in Israel's sense of encirclement), making them subject to the same less-than-professional leadership as the Egyptian military. The military was still more than capable, as it was better trained than other Arab armies and maintained enough autonomy to still be a strong force. However, they ultimately went along with Egyptian desires to spread forces out along the West Bank lines with Israel in case of war, rather that concentrate in Jerusalem or another strategic zone. This was a critical mistake that only one Jordanian officer opposed out loud, but Jordan's reluctance to even fight unless it had a strong sense of victory and/or a sense of strong public pressure to fight (which it did get, due to Egyptian reports that were false following the Israeli preemptive strike) meant it was acting more likely out of being forced by Egypt rather than out of clear strategic thought. Put that all together, and it becomes clearer why the Israeli side could win. Still, the part of surprise was quite important especially in the speed and decisiveness of the victory from the outset.

How did they have the element of surprise when tensions were running high and Egypt intentionally acted in a way the Israelis stated would cause war?

Egypt, frankly, didn't believe Israel would actually launch its own strike as it escalated. However, it did eventually expect a strike, roughly around when it happened. Its forces were mobilized, and it even told Jordan on June 4 that an Israeli attack would come within 48 hours (a correct claim).

However, the surprise was not down to Egypt not expecting a strike. Instead, the surprise was that Egypt was vastly overconfident in its ability to rebound from any strike, defend against one, and predict its path. That overconfidence was more costly to Egypt than one might think, since Egypt was expecting a strike after all.

Israel was, however, doing everything it could to make Egypt think the strike would come later. If Egypt's reaction time was off by mere hours, that would make a huge difference in terms of what Israel could destroy. Thus Israel made several probes with its air force to make Egypt think the attack would come in the southern Sinai, rather than from the Mediterranean. Thus the Israeli planes' flight path, out into the Mediterranean Sea and then circling back south into Egypt, and then back east, caught the Egyptian forces off guard. The planes flew incredibly low to avoid radar, and communicated only by hand signals so radio would not pose any vulnerability. Nevertheless, Jordan actually managed to pick up the Israeli planes on radar with a British-supplied radar facility. They radioed the Egyptians repeatedly, but the code for attack was ignored, delayed by the bureaucratic layers, and failed to filter down to Egyptian military forces out in the field in time, something Israel likely counted on. If the Egyptians picked up the signals themselves, that would be far worse. But the fact that it came from Jordan and had to work its way through Egyptian military command from that route was far slower and less reliable, it seems.

Continued in a comment below, responding to myself.