Do Balrogs Have Wings?

In the beginning, when the greater spirits went down to Arda, they could walk unclothed. If they had appearance at all, it was of their own choosing. It is debatable as to whether or not those choices were fixed when middle earth was sundered from Valinor and the west.
Personally, I believe that the form a Balrog chose was a reflection of their nature and was therefore full of fire and darkness. As you well may know, Melkor often strode the halls of middle earth in whatever form he chose. He is, however, much greater than a mere Balrog. Only at the end did he continually use the dark terrible form he is best known for and even then - it was most likely still his choice.

Balrogs have wings if it suits them. As pure spirits, they can most certainly fly but have no need for wings.