The two Forsaken: my explanation has always been the Dreams. We know that the citizens of Illian and Tear have horrific dreams due to the presence of Sammael and Be'lal. In fact, we know that Tairens actually saw an old man in their dreams.

Illianers probably saw a man with a scar. These two features make these two Forsaken distinguishable enough that one can explain Moraine's recognition by saying she read about these faetures at Adeleas and Vandene's house.

As for the Balefire, either the sisters knew something of it, or they had a book discussing it, enough so Moiraine could figure out the rest. It is easy to forget with the Wondergirls being around, but Moiraine, Siuan and Elaida are very skilled channelers too.

As for Asmo, it is easy. Moiraine regularly eavesdropped on Rand, and he made many a slip.