Or were there any plans to do so, for that matter?
The biggest issue was Stalin. At Casablanca Roosevelt and Churchill agreed to try to bring Turkey into the war on the side of the allies. However, Turkey was not in a position to go to war with Germany. Turkey had impressive manpower but didn't have the equipment. To bring them up to strength the US would have had to manufacture equipment for approximately 45 divisions. But arming 45 Turkish divisions would have significantly delayed opening the second front in Europe.
As for just using Turkey as a base to launch an invasion with US and British troops the problem of Turkish neutrality was added to the problem of Stalin's demand for a second front in Europe. Turkey was leaning toward the allies most of the war but there is a huge difference between leaning and allowing half a million troops on your soil.
Churchill expended a lot of diplomatic effort trying to get Turkey to declare war on Germany; presumably he had a plan for what to do if they had done so earlier. That said, the Balkans don't seem like a very promising place to open another front; at best they'd be tying down a German army, something like the Italian front. Not trivial, but not war-winning either. That's why the Americans were very insistent that Churchill not be allowed to, as they saw it, go haring off into the Balkans, but stick to the main attack.
Probably because it was a Neutral country. I'm assuming you meant during World War 2
Why would they? It was further from supplies than Greece or Italy. Further from Germany than either. What advantage did you see?