Good analysis, expect the point I mentioned above. Since a century, our doctrine is to train and fight on 3 fronts. East front. West front. And within our borders. Their is a reason why our Gendarmerie is the forth Force. They have the size, equipment and training level like a regular army.
So you think Turkey could go to war with Russia in the South Caucasus, Syria in the South, Greek in the Aegean at the same time? As you know there is a difference between a UDR (Ulster Defence Regiment) and the regular British army. The UDR were like Gendarmerie and could control internal operations, however they couldn't move around and be deployed like regular troops. In the case of the Caucasus you need offensive deploy-able forces, with armour and logistical support, air support. Specific training and operational experience within the office corps and so on.
If I look at Turkey right now, I see a rising regional power, getting used to its new position and trying to understand how to uses its power. Its being pulled in many different directions all at once and is a reactive power, not yet in control of its own sphere. But firmly in control over its own territory and internal politics. You have a strong ruling class, a strengthening culture and are trying to project that into overall geo-strategic security and control.
If I was Turkey there are three area's that are vital, the Northern Corridor basically Iraq and Syria, gaining a stable northern corridor is vital for Turkey, as that instability bleeds into internal Turkish security and politics, with the Kurds. Turkey has started moving into the North Corridor, but it has somewhat been blocked by the Russians and it also doesn't have a grand vision for the region to its south. As moving into that area means coming into conflict with Iran/Israel and Saudi, and as of right now Russia. Turkey needs to become the peace maker in the region. Flip Damascus and Baghdad from Iran and Russia and towards Turkey. For me right now this is the most important area for Turkey.
Then the South Caucasus, which is all about blocking Russian expansion from the North Caucasus. You did a good job defeating Armenia with your ally, but you need to get the Russians out and that conflict just further entrenched the Russian position. You can either counter-balance Russian actions with your own actions in the North Caucasus, but that's a risky position. Or you come to a compromise with the Russians, you know the compromise won't last, but it gives you time to strengthen local actors and strengthen your own position. Like the pipeline from the Caspian sea.
In the third area, the Aegean sea, your are hemmed in by Greece/France/Italy/US, so I would suggest a short navel victory over Greece and then a compromise on the boundaries, mineral resources and other issues with Greece. You want to show your economic and military superiority, with also showing your cultural and geo-political awareness of your own position. You will be a fair great power in the region, not a bully. Only after that can Turkey really turn to Cyprus which is the big fish in the eastern med. A key thing on Cyprus is getting the British bases off the islands, with that you can't blockade it or take it or ally in real positive and negative pressure to the local actors and government.
Once you have integrated Cyprus into Turkey, settled things in the North Corridor, South Caucasus, Aegean. Then you can start becoming more of a aggressive power and throwing your weight around, telling Israel what to do, moving into the Balkans and North Africa. After which the North Caucasus, Carpathian mountains and Black sea all come into play. And only after all this is Turkey geo-political and geo-strategically secured within its initial and wider sphere.