The year that the Gregorian calendar was adopted (1582), the discrepancy between the two calendars was at 10 days lag. So Christmas on Dec 25th of the old Julian calendar would have corresponded to Jan 4th on the new Gregorian calendar. So does that mean eastern orthodoxy celebrated Christmas on Jan 4th (gregorian) in the 1500s?
And as over time this discrepancy grew, did the eastern orthodox christmas correspond to different gregorian dates? At first, it would have been Jan 4th, then Jan 5th sometime in the 1700s, then Jan 6th sometime in the 1800s, and finally, Jan 7th now because the discrepancy is now at 13 days. And in the year 2100, the discrepancy will be at 14 days, so will eastern orthodox christmas then be celebrated on gregorian Jan 8th?
TLDR - why, historically, has the Julian date Christmas always fallen on Gregorian Jan 7 if the divergence in time between the two calendars was less than 13 days in the past?
The main answer is yes, the discrepancy between the Julian and Gregorian calendars increases by one day every 128 years. They were last in synch in 325 BCE CE, and the Gregorian calendar was designed so that they should be in synch at that date (the date when the formula for Easter was adopted).
So in regard to this part of your question --
At first, it would have been Jan 4th, then Jan 5th sometime in the 1700s, then Jan 6th sometime in the 1800s, and finally, Jan 7th now because the discrepancy is now at 13 days. And in the year 2100, the discrepancy will be at 14 days, so will eastern orthodox christmas then be celebrated on gregorian Jan 8th?
Yes, this is correct.
Given that that's exactly what you're questioning, I'm not sure why you repeatedly state that
the Eastern Orthodox Christmas always fall[s] on the Gregorian Jan 7th
because that isn't the case.
From 1800 to 1899, 25 December Julian corresponded to 6 January Gregorian. From 1700 to 1799 it was January 5, and so on. Here's an online converter if you want to play around. Strictly speaking they last coincided in 299 CE, but of course that's a long time before the Gregorian calendar was devised: in practice, for all dates prior to 1582 everyone uses the Julian calendar.
(Are you maybe thinking of the fact that the Armenian Church celebrates Christmas on 6 January? Because that's an unrelated thing. The Armenian Church has used the Gregorian calendar since the early 1900s, anyway. They've always observed Christmas on 6 January according to whichever of the two calendars they were using at the time, since at least the 4th century. That means that in 1900, for example, Armenian Christmas fell on 19 January Gregorian.)