Lebanon and Syria are 500 miles from Greece, just within Tante Ju and Canguro range.
In OTL, the Vichy French allowed Germans to land in their territory to ferry Germans to Iraq.
WI they had allowed the 15000 airborne troops destined for Crete (+ the Italian Folgore), to land in Lebanon and Syria in May/June 1941?
This would preempt the Allied/Free French takeover of OTL, and with the 8th Army already starting to reel from Rommel, would put a rocket up them, as well as stirring up the pro-Axis majorities in Egypt, Turkey, Lebanon, Palestine, Iraq and Persia.
Especially if the airborne drop coincided exactly with Barbarossa, Allied morale would take a dive. Some seaborne Germans might make it as well if they hugged the Turkish coast.
More useful than a drop on Cyprus, would be a probable suicide mission drop on Haifa. But it could render the port unusable to the RN and destroy the Haifa oil refinery and terminal and essential Iraq-Palestine oil pipeline, and divert essential Australian troops away from the Western Desert.
Haganah, Palmach and Irgun fighters vs Fallschirmtruppen: no quarter would be given!
The con would be that the Allied forces on Crete would remain untouched.

How would the Axis forces in the Middle East be reinforced and supplied? The Italian navy was no match for the British in the Mediterranean and you even admit that the Axis air fleets would be stretched to their limits in just getting paratroops to Lebanon and Syria. Turkey wouldn't be allowing troops or supplies through their territory so the Axis would have to rely on air and sea power here. Without Crete the Allied air forces could devastate the German transports, which at this range wouldn't be able to deviate from a near direct line approach to their targets. Even if the Axis gained control of Lebanon and western Syria they would still be hundreds of miles away from Egypt or Iraq. While the local pro-Axis majorities could help somewhat they wouldn't be able to lend much support to the Axis forces. At best they could provide a nuissance for the Allies.

Worst of all for Germany, the moment they attack French territory in Lebanon and Syria they essentially declare war on France and now have to fight a two front war just as they are putting all their energies into Barbarossa. While the French weren't a major force at this time it would still take a lot of troops to occupy Vichy France and this would simply give the Allies a free reign to occupy French North Africa and assault Italian Libya from the west.

All in all, not a plan I would recommend to the Wehrmacht, were I asked.

Why not do it before Barbarossa. Then you might ask for help from Stalin, who might "secretly" take care of the supply and troop movements through the USSR. Of course you´d have to give something in return, but I´m sure since already the eastern Europe had been cut to pieces a solution would be found.

This was also discussed earlier, and can be found with search mode.

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There's an option I hadn't considered. However, with Hitler I just don't see that as an option. By 1941 he was deadset on attacking Russia, not further dealing with it. Russia was making demands on Germany by then, specifically in the Balkans, and Hitler was having none of that. Furthermore, Hitler always pathologically hated the USSR while he at the same time always dreamed of having an alliance with Britain. To continue to befriend his Russian enemy that he was about to destroy in order to further afflict the British Empire that he admired wasn't a likely option.