Which Chinese dynasty built the majority of the Great Wall as it stands today?

The Ming, mostly between the fourteenth and seventeenth centuries. The stone-faced, watchtowered wall in every photograph is theirs; the earlier walls attributed to the first emperor were rammed earth, on different alignments, largely gone.

It also does not really work as a wall. Its purpose was less to be impassable than to be expensive to cross with horses and impossible to cross quickly with loot, and to serve as a signalling network - beacon towers could pass warning of a raid faster than a rider. Where it failed it usually failed politically: in 1644 a Ming general opened the gate at Shanhaiguan and let the Manchus through, which ended the dynasty that built it.

The claim that it is visible from space is false and unusually persistent. It is long but only a few metres wide, and at orbital altitude a few metres wide is nothing - astronauts have repeatedly said they could not see it, including Yang Liwei, China's first, which caused some awkwardness with textbooks at home. Roads and airports are far easier to pick out.