There need be no mention of Gaza in Genesis for it to have existed
before Genesis was composed, or for it to have existed in the time
Abraham is believed to have lived. The city evidently predates the
Amarna age and the reign of Pharaoh Thutmose III. So the non-mention
in Genesis is for you merely an argumentum ex silentio. The same is
true in the secular material for the Qadesh and Shur located south of
Hebron and Beer-Sheba.
There is only one "biblical" city named Beer-Sheba attested in the
biblical sources. A different post-biblical city with that name is
mentioned by Josephus as being located in lower Galilee (BJ iii.3.1
[39]). This ancient author at least did not know several Beer-Shebas.
The name clearly did not mean, in patriarchal times, any place well
water was found through providence, else every well dug by the
Patriarchs should have this name as the act of finding water in the
desert was hardly something that would have been defined as luck. It
is much simpler to see that Hagar's "wilderness of Beer-
Sheba" (whatever the name of the place was at this moment) became
popular overnight once news spread that water was there. Abraham
later gave it the name Beer-Sheba which stuck, according to Genesis,
and the rest is Jewish history.
Not everyone agrees that classic Philistines are being described in
the Patriarchal narratives. How about Proto-Philistines?
By Sodom, I meant the region of Hebron. Sorry for that. In any case,
my slip notwithstanding, there is certainly nothing in our text about
Abraham journeying to Ai where he then viewed the destruction of
Sodom from higher ground. After the destruction he left the area
north of Hebron where Mamre is located, headed south and passed
through the "south country" en route to the grazing territory of
Gerar closer to the coast, probably stopping (dwelling) somewhere,
but briefly, between Qadesh and Shur before actually approaching
Gerar. The tradition is infinitely more plausible then what you have
been attempting here on linguistic grounds.