How do two parties put together the AUTH data in the SKEME protocol?
how does $A$ know $g^y$ to include in the formula above?


Actually, what they mean here is, in fact, $g^y \bmod p$; that is, the value that $A$ received.  It wouldn't work to insert the literal values $g^x$ and $g^y$; apart from the fact that $A$ doesn't know the second one, there's also the practical difficulty that since $x$ and $y$ are (perhaps) 256 bit random values, the bit representation of $g^x$ and $g^y$ may consist of circa $2^{256}$ bits in length; rather too long to process.

In general, when the group we're working in is understood, we often write $g^y$, with the fact that we're talking about operations in the group, and not as an integer, being implied.