The treasure ship specs on wikipedia are unbelievable for a wooden ocean going vessel. Maybe something that big could have been used on rivers, but there is simply no way that a wooden ocean going ship of that size could have sailed to East Africa. It is listed as 415 feet (125 meters!) long by 170 feet (52 m) wide. That is about half the size of the HMS Titanic.
It was more than twice as big as the Santissima Trinidad, the largest ocean-going warship ever constructed out of wood. And that ship was barely even sea-worthy after it gained a fourth deck.
I have no doubt that the late Ming Empire was technologically advanced, in fact they were probably the richest and most advanced civilization on earth at the time. However, I simply don't think it is possible to built a wooden vessel around the size of the Great Eastern and sail it around the world's oceans. No ocean going vessel of the listed size of the Treasure ships existed until the 1860's, and those incorporated steel in their construction, and were powered by screws.
The ships managed to reach East Africa in the fairly brief time that Zheng He and company were given their full go-ahead.
I don't know how much of a stretch from that it could be for the ships to just hug the coast around Africa down to the Cape of Good Hope and make a right. (Okay, the Cape of Good Hope might have been something of a challenge as per what you wrote, but, given determination, a dash of luck, it could be done, no?)
I'm not saying that Zheng He and company would have gone across the Atlantic and Pacific oceans, but, especially given time, sticking close to the coast and working their way up to Iberia and the Mediteranean doesn't seem too fantastic.