Is Turkey as relevant to NATO as it was during the cold war?
Just doing one of the questions: "Is Turkey as relevant to Nato as it was during the cold war"
!Total rewrite following Denis de Bernardy's comment!
Necessary context for the comparison you want: Nato was originally formed as opposed to the Soviet Union, which has fallen, so Nato itself is not as relevant as it was in the cold war. Nato is now hanging on to its existence and currently has Russia as #1 threat (until China gets up the speed) and a general interest in stability. Russia is not the big deal that the USSR was.
Turkey's relevance to Nato Now is not primarily its ability to affect the outcome of the war in Syria . Or the fact that it houses 3 million refugees that it could "dump in Hamburg at will, or that it is intervening in Syria by attacking the Kurds.
I consider Turkey's ambition in Syria (and Iraq) to be limited to disabling the Kurds who have in the past sponsored terrorism in Turkey as part of their ambition to create a Kurdish state. Although Turkey is mostly Sunni they are not hyper-religious, so I cannot see an ideological confrontation with Iran. This is Nato-neutral.
Turkey's relationship with Russia is a Nato concern, as indicated by the linked The American Conservative article (BTW, TAC is mostly excellent imo). Turkey's purchase of sophisticated Russian missiles is interesting. Why would they need these? Do they have Hegemonic ambitions in the Levant? Do they have a Bomb project? There has been talk, and if true, NATO needs to know.
Turkey's fundamental value to Nato is that Turkey controls the Bosphorus, the link between the Black Sea and the Mediterranean. You must look at this map, and then reduce the scale to see the significance.