I'm going to address the sub-questions, then I'll move forward to the main question.

Although Israel was threatend by the aggressions of the neighboring countries, the war began with Israeli initiative, even by it's own admission.

This much is true. However, Israel was still viewed as the victim. The first reason is that they had waited over two weeks before striking, while being blockaded. Israel had not only been in constant communication with world powers about what it would do if this crucial waterway remained closed to shipping bound for Israel, it had warned that such a closure would lead to war 10 years before, in 1957, at the UN General Assembly in front of the entire world's representatives. Israel had been quite clear, and the blockade definitely played a large part in how states viewed the ensuing conflict. In 1957, Golda Meir (at the time, Israel's Foreign Minister) had declared:

Interference, by armed force, with ships of Israeli flag exercising free and innocent passage in the Gulf of Aqaba and through the Straits of Tiran will be regarded by Israel as an attack entitling it to exercise its inherent right of self-defence under Article 51 of the Charter and to take all such measures as are necessary to ensure the free and innocent passage of its ships in the Gulf and in the Straits.

This was not lost on the world, and especially not on the US, which was in communication constantly with the Soviets (who were allied with Egypt). Even had the US forgotten, Israel went to great lengths to remind the US that it would act if the world did not reopen the Straits. The US had said at the time that it found Israel's expectations of the Straits staying open lest war occur "not unreasonable". In a memorandum prepared by Special Assistant to the President Rostow on June 2, 1967, recording a conversation with Ephraim Evron, who was then the Minister at the Israeli Embassy in DC, Evron had asked the fundamental question:

Do we still stand by Lodge’s assent (and Foster Dulles’) to Golda Meir’s statement in General Assembly? The track discussed between Eban and the President—on which we have hitherto been moving—is consistent with the commitments made in 1957 that we would ourselves assert the right of innocent passage; that we would assert that right on behalf of others; but that we would have to engage through our constitutional processes if we were to use force to assert that right with force on behalf of others. What is involved in the track he is suggesting is reaffirmation of the other branch of our 1957 commitment incorporated in the Golda Meir statement and Lodge’s assent.

In this light, and in light of the other events leading up to the war, such as the remilitarization of the Sinai and lining up of over 100,000 Egyptian troops on Israel's border (as well as Syrian troops in the north), the expulsion of UNEF peacekeepers, and the fact that Israel waited with reserves mobilized and a blockade on the path that it received 90 percent of its oil through, world powers saw Israel starting the war as...less than awful. The US did not want it, nor did the Soviets, but countries were far more understanding than they might've been had it been a repeat of the Suez, for example.

I would expect that in a post World War II frame of diplomacy, the US and Europe wouldn't see an expansionist state in the middle east in a positive light, especially considering the pressure from the USSR.

Part of the reason for the sympathy for the Israeli position was the aforementioned scenario in the weeks leading up to the war. At the same time, the US did not view Israel as "expansionist". They knew that it held territory, but viewed its holding as the result of war, and did not think Israel intended to keep all of the territory. Indeed, Israel communicated to the US precisely that.

The goal of the post-war situation was to try and use it to force a broader peace deal between Israel and its neighbors. There was hope that now, having had such great losses, a deal could be forged with the chastened Arab states. In the initial days following the war, positions were taken down: Arab states demanded full and unconditional withdrawal from all territories Israel had gained in the war, before considering any peace or negotiations. Israel's response was that it was willing to give up the Sinai and Golan Heights, but wanted to keep the West Bank in a peace deal, and would certainly not cede them before negotiations.

The Soviets, of course, did accuse Israel of being expansionist. Their position mirrored the Arab position: Israel must withdraw from all territories unconditionally, and in exchange there would still be no peace, but the Soviets consented to submitting the blockade of the Straits to the ICJ. The UN General Assembly was convened at the USSR's request, and resolutions that even said that full withdrawal should be coupled with recognition of the right of all states to live in peace and security were rejected by the Arab states. What resulted, of course, was UN Security Council Resolution 242, the result of negotiations between the US and USSR when even the UN General Assembly could not pass a resolution conclusively.

Walt Rostow, the Special Assistant I mentioned earlier, suggested "lean[ing] against [the Israelis] just enough to keep their thinking from becoming too quickly set in the concrete of their current extended territorial possessions." They pushed Israel to be "flexible, patient, discreet and generous", pushing for a resolution that acknowledged that territory is not admissible by war but also that the final resolution of the conflict would not require Israeli withdrawal from all territories, or the territories. Indeed, in what is a rather unique resolution historically, the English draft intentionally called for peace in exchange for a withdrawal from "territories" acquired, which the resolution's authors explicitly acknowledged was intentional. As put by Lord Caradon, the UK's representative to the UN:

We all knew – that the boundaries of '67 were not drawn as permanent frontiers, they were a cease-fire line of a couple of decades earlier . . . We did not say that the '67 boundaries must be forever; it would be insanity.

While these interpretations were not uniformly accepted, Israel and Jordan both agreed to the resolution. Nasser, while telling his generals that "what was taken by force will be retaken by force", suggested privately that he might be willing to discuss peace with Israel. Iraq and Syria rejected the resolution outright, as did the PLO.

Ultimately, these were not viewed as the actions of an expansionist state by anyone besides the USSR and its spheres of influence. In the West, Israel was viewed as both the victim of a war it did not want and also willing to return some of the territory, if not nearly all of it, for the sake of peace.

Israel also made another smart political move with its communications in the days following the war, which also helped it argue it was not seeking expansion. It made a June 19 decision in the Cabinet to exchange the Sinai and Golan Heights for peace with Egypt and Syria. This separated out those territories not part of the British Mandate, which meant that Israel was offering to return to Egypt and Syria the territory that no one disputed was theirs (Egypt's occupation of Gaza was not viewed as legitimate in the world community), in exchange for peace. Thus Israel was viewed as less expansionist; it argued that the Mandate's former territories were never Egypt's and Syria's, and therefore shouldn't be part of any peace deal with them. For Jordan, there was no real consensus on what to do with the West Bank. There were debates about an autonomous Palestinian area or even state (joined with Gaza), or a return of some of the West Bank. This offer was certainly communicated to the US, which helped Israel's image. What's unclear is if it was ever communicated to the Arab states. Abba Eban, Israel's Foreign Minister in 1967, insisted that the offer was. However, historian Avi Raz disputes this, saying the resolution was meant to influence the US view of Israel, and was never passed to the Arab states at all.

Continued in a reply to my own comment.