You're talking about Jewish paramilitary activity in Palestine as though it were one, unified group. It wasn't.

There were huge splits over strategy between the Hagannah and the Irgun. The Irgun wanted all of their demands, in full, immediately, (given that many of their leadership were holocaust survivors this was more reasonable than it sounds) and the British stood in the way of that with immigration restrictions and partition plans, among other things. For a first hand account of this thinking read Menachem Begin's "The Revolt." It's obviously biased, but that's sort of the point if you want to understand what the Irgun was thinking.

As for the British stance towards Israel, it's more nuanced than what you're describing. For one thing, a good amount of pro-Zionism in the British government was actually anti-Semitic. Balfour and Lloyd George apparently thought that Jews had an enormous world-wide influence, and so favored Zionists like Chaim Weizmann to placate them. British policy in Palestine also favored the creation of a state that wouldn't have looked anything like modern Israel. Read the British white papers on Palestine, or the book "One Palestine, Complete."

I don't know anything about the international relations of Egypt under Sadat, sorry.

I asked a similar question a few weeks ago, and got a great response that you may find helpful

What exactly was the British stand towards Israel? They seem to have support the partition yet they then impose control on number of Jews Immigrant as well as saying things like :"we don't take orders from jews" (sir john shaw). I thought the British had supported Israel from the beginning (Zionists.. WW1+2)?

Well, the British walked a fine line as far as support for the Jewish state. The Mandate they were granted was supposed to give self-determination to the peoples of the region. The British regarded Palestine as a "special case", which they also separated as different from Transjordan (which had also been considered Palestine). British controls on Jewish immigration were intended (if you're talking about the White Paper of 1939, for example) to placate the Arabs who had risen in the 1936-1939 Arab Revolt. It even stated that an Arab state, if conditions were favorable, would be established in 10 years. However, these policies were not implemented, partially due to Arab rejection and partially due to continuing support for a Jewish state, which the British never really stopped committing themselves to. However, the British were growing weary and also scared of rising tensions in Europe, and wanted to defuse any possible escalation (though the Revolt had largely petered out by 1939). It was intended to shore up Arab support in the coming war, by most accounts.

It's also important to note that different British governments had different stances on the issue. When Prime Minister Clement Atlee's government took over in 1945, for example, it was usually more inclined to support the Arabs than previous governments like that of Churchill, who noted the support for the Jews in 1944 might wane if groups like Lehi didn't stop assassinations like that of Lord Moyne.

Also, Why did the jews launch the attack at the King David Hotel against the British servicemen when the British is one of the very few groups that do support them in the beginning? why would they do this when it was the british who helped them migrated to Palestine?? why would the extremist groups attack the 'closest' thing they have to an 'ally' especially when the Jews are developing tensions with the surrounding arab states? What was the point of the attack??

These extremist groups, mainly Lehi and Irgun (though Haganah was guilty of a few attacks, like that on the Semiramis Hotel) were motivated primarily by a feeling that diplomacy would not work. They viewed the British as equally obstructing their path to a state, and viewed their promises as disingenuous. They were also heavily influenced by the Revisionist Zionist cause that they worked from, touted by Vladimir Jabotinsky. Jabotinsky argued that the attempt to enter Palestine would only be solved by fighting, and that the Arabs would not submit, and so they had to be made to submit because the Jews had more urgent need for a state than the Arabs of the area. This ideology, when viewed through Irgun's and Lehi's lens (Lehi was influenced by even more extremist ideology), saw the British as yet another obstacle that could only be overcome by violence. They didn't see diplomacy as working, or the British as an ally, so they were more willing to attack the British.

Haganah, the largest Israeli paramilitary group (and the one that formed the backbone of the IDF and stayed intact after the war of 1948) was less inclined to take this tack, though it did so by attacking British strategic and military targets (for the most part) nearer to the establishment of Israel. By then tensions had run so high and feelings of distrust even higher, that there was quite a lot of fighting going on between all groups.

Lastly, after Sadat came to power; it says he want to seek a detente with USA as he removed all Russian advisors, yet he still decides to engage in war with Israel at the same time? Why did he have this policy.. I don't see why it would make sense to seek friendly relations with America while trying to invade Israel at the same time.. Wouldn't it benefit him more if he had stayed with the USSR?

Sadat had indeed attempted to engage with the United States. However, Sadat's overtures were not always replicated. Sadat was not taken seriously by Kissinger, and was constantly saying that he wanted either peace or war (if memory serves correctly). Kissinger himself said, in an interview, that he did not take Sadat seriously until after the war (interview conducted in later years, of course). Sadat therefore saw an opportunity to attempt to reclaim lands lost in 1967 (as well as honor) and saw no recourse by which he might do so otherwise. Peace talks were fairly nonexistent, and Sadat felt that he was not being taken seriously enough as a major power. When the opportunity to start the war arose, then, and Assad and Sadat were able to keep it a secret (meeting in clubs to discuss plans, making the Israelis think the movements were just exercises, the Israelis blundering greatly to not understand what was going on as well), he saw it as a better option than not getting any peace at all. After the war, and a few more years of hostility (during which he was taken more seriously land at least two Sinai disengagement plans were negotiated through Kissinger's shuttle diplomacy), he was able to foster better relations with the US and Israel, good enough to get over his personal hatred of the Israeli PM at the end of the '70s (Menachem Begin) and be taken as a serious player in the Middle East. He was taken seriously, Egypt regained some of its honor and all the land it lost besides Gaza (which was occupied anyways by them post-1948), and it was able to sign a peace with a foe who had nuclear capabilities and a clear conventional advantage in most conflict (thanks especially to strengthening US relations). And he got better ties with the US, as I mentioned, which meant better arms deal and aid.

Also, in the peace talks between Israel- Palestine.. Which of the peace talk do you think is the most significant one, and why?

There haven't really been peace talks with the Palestinians that don't fall within the last 20 years, besides Oslo and the Madrid Conference (which went nowhere, but I guess was significant in that Palestinians faced Israelis). I guess you could argue that Oslo is the most important, specifically because it reaffirmed the Palestinian renunciation of terrorism and the recognition of the Israeli state, as well as Israel recognizing the PLO as the representative of the Palestinians. It also provided the basis for later talks, though none too well.

Other than those, if you're including the wider conflict outside of the past 20 years (since that rule of the sub), then the Camp David Accords definitely. It attempted to create a significant timetable for Palestinian autonomy, but more importantly it drastically changed the balance of the Middle East. It removed the most powerful and populated Arab country from the conflict (at least, so long as Egypt agreed to continue the treaty) and also provided the background for later land-for-peace deals being executed (or well, tried). Camp David was historic: for the first time, an Arab country had negotiated with Israel and come to a peace agreement, in defiance of the 3 no's of the Khartoum Conference (no recognition, no negotiation, no peace, I believe). Jordan would follow as well, but Egypt making the first move was extremely notable, and it altered relations even amongst the Arab nations. Sadat would be assassinated, the Arab League would vote to move its meeting place out of Egypt, and Syria's president was furious. Definitely changed a lot, that's for sure!