Turkish infrastructure will limit the logistics of any German ground operation mounted through Turkey anyway, as we discussed earlier. The force they could send down into Syria and Iraq would be pretty modest, 10-12 divisions max.
Kinda? As I noted earlier, the British staff plan assumed that the core of the assault force would be by 4 panzer divisions, which were the main offensive strike force of the German army and who did the bulk of the slicing, dicing, and encircling in Barbarossa. This would be on top of the 2 already dispatched to Libya, OTL to act as defense and IATL presumably to act as the anvil to the Turkish forces hammer. That's 6 panzer divisions total at a time when Germany had a grand total of 21. The Germans used 17 panzer divisions for Barbarossa, so we're looking at dropping a minimum of two panzer divisions from the Barbarossa thrust. Given how thin the German achievements in Barbarossa were managed historically, that could result in major impacts on Barbarossa itself.
The North Africa campaign used about an Army Group worth of trucks for a poorly-equipped panzer corps, because they were operating entirely too far from railheads.
The Germans used around 600,000 trucks in the invasion of the USSR, which is 200,000 trucks per army group. For Africa, there were 2-3,000 trucks in each of the two (later three) panzer divisions plus an additional 5,000 (give or take a few thousand) in the non-divisional elements that did the bulk of the long-range haulage. That's about 14,000 (later 17,000) trucks throughout the entire corps... call it 20,000 at most, plus whatever motor-transport the Italian units had. So while the truck requirements for the single panzer corps was disproportionate, it wasn't to the degree of an entire army group's worth.
It's also technically incorrect to say that they were operating too far from railheads, but only because there were no significant railways in Libya for them to have a railhead out of so the main measurement of distance was to the nearest seaport, but that's terminological quibbling.
Assuming anyone pays attention to the German logistics types, they'll have the same insane truck requirements for a trans-Turkey operation out of Beiruit or wherever ... trucks that will therefore not be available for Russia.
While undoubtedly there would still be demands, they won't need so many trucks as to haul across the
entirety of Turkey because, unlike Libya, Turkey DOES have a network of railways in 1941.
As one can also gleen from the above map, the rail lines do continue down into Syria and Palestine along the coastal region, so if anything the truck requirement for this Turkish operation would be less then for the North African Operation. Requirements can be lessened further if the Germans manage to capture or neutralize Cyprus, which would open up the option of secure coastal shipping along the many reasonably developed ports.