Adam, all of your sleuthing and deductions are amazing.  Needless to say, I have been absolutely hopeless at deciphering the actual images in JRI-Poland (from AGAD) but your point about Gittel Kurz' hometown being specified whereas Chaya Rost's was not did tend to make me believe they were two different women .  On the other hand,  there are 5 birth records with Gittel Kurz as mother and only the single death record in which Chaya Rost's name appears so now t  am tempted to believe that they are the same woman and that the Gittel listed as Sossie's mother on the 1875 birth record is the same woman as the Chaya Rost who is listed on Sosie's death record in 1884 (though of course the child is listed as being exactly 7 years and 2 months which should have put her birth in 1877 and not 1875.)  Interestingly on Sossie's birth record under her mother's name either Rost or Rose is crossed out.  Of course, this could just have been the name of the midwife put in the wrong spot, crossed out and moved to the next column.

I have no knowledge of any relatives earlier than my great grandfather Chaim.  So I have no idea if he had a sister Jennie I have only been able to find one sibling for him, his brother Frank (or Favish) with whom he apparently did not get on but I am only just beginning my research on this part of my family so others may turn up.    And in all honesty, it probably was from JRI-Poland that I deduced my great great grandparents' names. The symmetry of Josef Isaac/Alter's father parents being  Chaim Elio and Sossie Kern which were in turn the names used for his children was pretty compelling. 

JRI-Poland did uncover a surprise for me when I discovered that Chaim Elias Kern and his wife Gittel Knecht had a boy named Usher in Zhitomir who died within 18 days.  (Usher was Gittel Knecht's father's name.)  On the boy's Zhitomir birth record only his mother was listed because I guess his parents were not considered "legally married" there.  On the boy's death in Chortkiv, however, both his parents are included and his father (my greatgrandfather) is listed as Chaim Eliasz Kern.  I am unable to read the column headings in many of these records but strangely on the boy's birth record in the 4th column from the right, the name Rost appears. So for the time being, at least, I am leaning to the opinion that all these Alter Kerns were the same guy.  Probably the bigger shock to me was that Kern was the family name in the Old Country since many family members have long believed it was an Americanization of Cohen.