1)What POD or PODs would be required for this to happen?
Hitler sends Rommel more Armored Divisions, and perhaps calls of Barbarossa to concentrate on the Mediterranean.
The Capture of Malta rather than Crete. This could cut off Allied supply lines and the Axis Dominate the Mediterranean.
2)Assuming Nazi victory in Egypt, what immediate affects would this have on both the Allied and Axis war effort?
The Allies would loose all access to the Mediterranean. Cyprus would be besieged by U-Boats and the Luftwaffe. German forces would have control over the Suez, allowing for a safer and quicker way for U-Boats to move into the Indian Ocean.
Germans would have a massive morale boost, and production would be very high. Hitler would start plans to attack Russia on two fronts, one from the Middle East, and the other from Eastern Europe.
The Allied war effort would probably begin to fail more and more. India would now somewhat be a direct threat from the Germans, The Holy Land would likely fall as well.
Pro-Axis Rebellions in Iraq are more likely to succeed, with the Germans only a few hundred kilometers away. Persia would probably be cooperative with the Germans as well.
3)How soon could the Nazi's start the invasion of the Middle East? What sort of opposition could the Allies muster?
If Hitler calls off Barbarossa to concentrate on Africa and the Middle East, here is a basic timeline.
March 1941- Hitler calls off aerial invasion of Crete, and instead plans for aerial invasion of Malta.
April 1941- Malta is captured by German air forces. Supply lines to Africa for the Germans are now safe.
May 1941- Hitler begins to send more troops to Rommel, who starts to make gains into Egypt.
June 1941- Allies are in full retreat to defend the Nile. Rommel as much more experienced men, and a lot more soldiers than the Allies by now.
July 1941- Rommel attacks the Nile and overwhems the Allied defenders, taking Alexandria and Cairo. The Allies being to form a defensive line at the Suez.
August 1941- Rommel attacks the Suez, eliminating the allied presence there, and prepares for an invasion of the Middle East.
The Allies would probably muster a lot more men in the Middle East, but most rushed out of training for the purpose of defending the Middle East.
The Germans are more likely to defend Africa from Invasion with far more men on the continent. Operation Torch a year later would fail or be called off.
4)What would happen if the Nazi's were to take Palestine?
Hitler would claim it was Germany's destiny. The Palestinians would likely be indifferent to the Nazi occupation. Maybe some would be incorporated into the SS' Muslim Divisions.
Huge propaganda victory, Hitler makes a visit to Jerusalem, and orders the burning of all major Jewish monuments.
5)How long would the entire conquest of the Middle East by the Nazi's take? Is an invasion of India possible?
Rommel could probably occupy the Middle East by the end of 1942 and attack Russia in a newly planned Barbarossa from Iran. The Germans would now have unlimited capabilities with most of the oil fields under their control.
An Invasion of India would likely take place. If Hitler promises Indian Independence, then you would see rebellion against the British with the Germans in Iran.
6)If the Allies are defeated in the Middle East, is their any way they could keep fighting the war?
Perhaps, Sealion is still impossible, if the British could hold onto India, which is unlikely with Japan likely to attack as well, then we could see a ceasefire. Hitler controls all of India and the Middle East, as well as Europe. Britain is now out of the war.
7) Did the Nazi's have any plans for what to do with the Arab population, both Muslims and Jews? What was the Nazi stance on Islam?
The Nazis actually respected Islam, but would likely kill all the Jews. Arabs were actually IOTL incorporated into several SS Units.