I'm not going to get into the latter part of the question, because I know that it's a question relating to the events of today, and I don't really think those are good to get into. The 20-year rule is definitely applicable there.

How did countries react to the 6-day war is a good question to ask, though!

The Arab nations were understandably embarrassed by the resounding defeat. They lost quite badly, and Syria lost the Golan Heights, Egypt the Sinai and Gaza Strip, and Jordan the West Bank. This could easily be regarded as the basis for the 1973 Yom Kippur War, as they wanted to "right the wrongs" of the loss in the past. Immediately following the loss in the war, they had passed the Khartoum Resolution, which specified that there would be no peace, no recognition, and no negotiations with Israel. This would effectively lead the door open to the 1973 War, and the Arab states remaining a united bloc in this regard is also a possible contributor. Not until 1979 would an Arab nation involved in 1967 make peace with Israel, and that was Egypt alone. It would face a huge outcry and condemnation from other Arab states for it. Egypt received the Sinai (though it had to be kept mostly demilitarized) in return for peace. Note that the Camp David Accords were 1978, but the peace treaty was signed in 1979. The next Arab state to make peace with Israel would be in 1994, and that was Jordan. Only in 2002 would the rest of the Arab states make an offer for peace.

In the global community, Israel gained a very large amount of recognition militarily, but what followed would be very difficult for it. The issue of refugees would contribute to a decrease in Israel's standing with the UN, especially considering UN resolutions like UN Resolution 242 which called for Israel to evacuate the occupied territories and called for a solution to the refugee problem. Nations recognized Israel's military might, and will to survive, but were reluctant to accept the status quo with regards to Israel's handling of the occupied territories and the refugee problem, both of which continued well past 1967.

The US's choice to begin providing military aid to Israel began right around the Six Day War. Because of the Soviet support for the Arab states, and the way it seemed they were aligning, the US began to lose hope that it would have any influence in the area. This prompted the sale of technology to Israel in 1968, and led to increased cooperation that would eventually lead to the situation we have today. Prior to 1967, there was very little support militarily from the United States, which was supplying military technology to the Arab States in the hopes of currying favor with them.

Despite the USS Liberty mishap, there is no indication that the United States was holding a grudge that overpowered its hopes of influence in the Middle East, and it would eventually agree (under Reagan) to make/maintain Israel as the strongest military power in the area. The 1967 War laid the foundation for many of the wars to follow, and many of the problems that persisted throughout the 20th century. I can't get more into specifics unless you let me know what else you'd like to learn about :).

Source:

Phantom Fracas: The 1968 American Sale of F-4 Aircraft to Israel David Rodman Middle Eastern Studies, Vol. 40, No. 6 (Nov., 2004), pp. 130-144