discovered by his grandson Menachem Goldstein while visiting his father in London.

From the front of the first book

At the bottom right, he twice writes in Hebrew what appears to be "Naftali Yitzhak Fradmin" - the aleph is clearly emphasized in the first instance. The name is misspelled both times and the vertical lines in the second instance makes me wonder if there isn't some kind of wordplay here..

To the left of those, he writes (in Latin letters) "I. Freedman" three times, which I assume is a reference to the same person.

I have no idea who this Freedman or Fradmin is.

R' Dovid Shapiro writes:
It's common to find lots of names on old seforim. Usually former owners, or even someone who borrowed it. You will notice that Naftali is spelled wrong in the Hebrew, both times. The first time Freedman is written Fradmin in the Hebrew. It's looks like it was written by a child, or at least someone who doesn't know Hebrew well. In the "English", I think the first initial is a European 'J', not I. This suggests that it was written on the continent, before the seforim arrived in England, and may well have been a former owner.

My son Avraham was born to me Rosh Hodesh Nisan 5676, in their count [=Gregorian calendar] April 23, 1916 [actually 3 April]. We hope to merit to raise him to Torah, the marriage canopy and good deeds.
My son Yaakov Yehoshua was born to me on the Tuesday of the week we read Parashat Toledot 5680, 18 November 1919, 25 MarHeshvan. We hope to merit to raise him to Torah, the marriage canopy and good deeds.
My daughter, the girl Esther, was born to me Motzaei Shabbat Bereishit 5685, 28 Tishrei, in their count October 25.
My daughter, the girl Chaya Sarah, was born to me on the Sunday of week we read Parashat Yitro, 16 Shebat 5686, 31 January 1926.
We hope to merit to raise them to Torah, the marriage canopy and good deeds.
Note that the third son, who died, is not listed here.
My father, my teacher, of blessed memory died in 5667 on the eighteenth of the month of Adar. May his soul be bound in life. 1998.
My mother, my teacher, of blessed memory died 5658 on erev Shavuot and was buried on the day of Shavuot, on Sunday. May her soul be bound in life. 1907
I was born 5626.
There are several problems with this death information. Let's assume he meant 1898, not 1998. That fits the Jewish year he lists for his mother, while 1907, which he wrotes for his mother, fits the Jewish year for his father. I am assuming the Gregorian years are correct because in 1898/5658 erev Shavuot was Shabbat which fits Sunday burial. In 1907/5667 erev Shavuot was a Thursday, so the Sunday burial would not be right. On the other hand,"Yom Rishon" may not mean Sunday, but rather the first day (of Shavuot). 

I have inquired in Tarcal and they say they have no Baum deaths in either 1898 or 1907.

I am very disappointed that he does not record the names of his parents' parents. I really hope that if I find death certificates, the parents' names will appear.
According to the 1867 census, he was two years old, so 5626 may not be precise, though my guess is that it is.
This is the frontspiece for the book in which all the above appears.

It is the books of Kings and Chronicles, published in Vilna in 5634/1874.

It includes the basic commentaries, in addition to a commentary in Yiddish.

Interesting here is the stamp at the top of the page:


Salamon was the next younger brother, who went to New York sometime in the 1880 or early 1890s. I do not think they ever saw each other again, though Salamon's two elder daughters visited London in the 1930s.