I'd play as Turkey and choose a Fascist dude as my first Minister, while pursuing the Collectivist National Focuses so as to quickly become Fascist. Afterwords I would go down the General Improvements Tree about Infrastructure/Industry so as to modernize Turkey. Later join the Axis, take Iraq, Syria, Palestine, maybe even get Greece before Italy does and see how far I get. After that I'm probably screwed with the Russians to the North, but hey maybe I can push and get to Baku and strangle Soviet industry to death (although they don't really need oil for just straight up infantry... meh).
If you join the Axis, Turkey becomes less of an opportunity and more of a threat to the Soviets, or at least their oil fields on the Csapian.
Army based on Infantry, Artillery, and Anti Tanks against the Soviets.
As for Turkey's plan, what the Turks might have done is always intriguing. A strong ground defense as you propose seems wise in the Caucasus and Turkish interior - an infantry+support force is convertible to opportunistic offense through the mountains if desired after Russia is otherwise heavily engaged. Presumably they will be well conditioned for rugged and freezing environments.
Light/Medium Tanks (Panzers as well as Turkish Tanks probably) and Motorized infantry in Africa and the Middle East. Air force of mostly Fighters, probably won't have the industry for anything else. Navy of subs and destroyers to attack British shipping in the Mediterranean, I'll let the Italians handle the surface fleets.
I expect Turkey has ample manpower relative to its industrial resources, so needs to economize on materiel and industrial capacity. For the mideast you might consider light armor, motorized support, and mounted infantry (aka cavalry) for efficiency in mobile operations, with firepower-heavy infantry divisions for sieges, pinning, and in difficult going.
The Turkish coastline is long but not very welcoming to an invader seeking to press inland, except at the Bosporus, where I expect fortifications will enhance the defensive positions so you can have a reduced garrison and field larger reserves elsewhere.