The purge happened in 1992, so if Alex was in fact sixteen, Ben would not have been able to have her since she was a baby, did he take her as his own after this, if so, where was she before hand and did Ben have a more active role in The Hostiles before the purge?
Could it be that the Hostiles/Natives took Alex during a time when Ben was quite active around both them and the Initiative (before the Purge however) and, therefore, while not being her father he would still have been around only to take the role of her father after the Purge.
There is definitely an issue with the timeline here. Born 1988, Purge was 1992, unless Alex was staying with the others when Ben was still with DHARMA being "Patient".
The elusive Radzinsky helped care for baby Alex after the Others kidnapped her (at their request) in the safety of the Swan station prior to the 1992 arrival of Inman and the Purge that followed. Shortly before his new co-worker arrived, he returned Alex to the Others, who cared for her until giving her to Ben after the Purge.
When Locke (fake Locke) and Ben return to Dharmaville, Locke leaves Ben to do an errand. The errand was to dig up Alex's body so that the Smoke Monster can inhabit the body and tell Ben to do whatever Fake Locke asks.
This seems to imply that fake Locke (now known almost conclusively as the Monster) would need an exposed body in order to inhabit it. However, Locke's body was in a coffin and then in a sealed crate. How is this explained?
It looks as if MiB is able to duplicate any unburied body, regardless of its being in a casket or not: Yemi's body was in a plane fuselage hanging dozens of feet in the air and this was no hindrance to MiB, so I can't see a casket being any less "penetrable".
When she appeared to Ben underneath the Temple that was actually the MIB trying to get Ben to completely follow Locke.