Do I have any control over the audio mix other than what's done on the desk?

Some Shure monitor systems, for example, the PSM300, feature MixMode® technology, which allows for 2 streams of audio to be sent to the receiver. This allows the user to mix between the two signals, making one louder or quieter. One good use of this process would be for instruments to be sent to one channel and the lead vocal to the other. This allows a singer to turn up their vocal if they need to hear more of it.

How does a wireless monitor system get connected to a mixing desk?

Floor monitors are generally connected to an auxiliary or monitor output to provide that specific monitor with its own mix. In-ear monitor systems basically replace the floor monitor on that output. A mix is sent to the in-ears in exactly the same way as to a floor monitor.