Mance must know quite a lot about Melisandre. But as for Reek, Mance and his 'washerwoman' have been keeping an eye on Theon, some of the women asking him how he captured Winterfell. Theon wonders if Abel (Mance) is looking for a way out of the fortress. But the answer to what Mance might know is to be found at the end of the chapter, A Ghost in Winterfell, just after Theon hears the heart tree saying "Theon" and then "Bran". Three of Mance's washerwomen come for him; the last words in that chapter are these:
Now while one of the things Mance wants to discuss with Theon is their escape (with 'Arya'), I can't believe that Mance doesn't ask for more information than what's required to make the plan work. He'll have asked about anything else that he thinks might be of use later.
Oh, and the "You want to die as Theon?" question makes it clear that Rowan is fully aware of Theon's identity issues.