The Book of Genesis says that Abraham was from Ur Kasdim. Usually English translations renders it "Ur of the Chaldees/Chaldeans". It is impossible for Kasdim to mean Chaldeans if the Torah was written by Moses since the Chaldeans only came to Mesopotamia around 8th century BCE. And Abraham cannot be Sumerian (Ur was a Sumerian city) since Sumerians were not semetic people. So what was Abraham's ethnicity?
This is two questions:
1) How could the Torah describe the city of Ur as being of the Chaldeans if they didn't exist yet?
The Torah clearly describes a city in which Avraham and his father resided as being called "Ur Kasdim", so either
a) The Torah is using a "borrowed" and contemporary-to-a-future-time name (Chaldeans) in place of some proto-Babylonian people (Sumerians?) - see here for a similar example:
b) There were in fact Chaldeans at the time, and the assumption in the question is mistaken.
2) What was Avraham's ethnicity - and were the inhabitants of Ur Semitic?
Avraham was certainly descended on the male line directly from Shem and then Ever - see the details of his descent at the end of parshas Noach - so definitely Semitic in the basic sense. The inhabitants of Shinar/Bavel/Iraq were likely descended from Cham (10:10 - though it's not conclusive - and I don't know if Ur is the same as ancient Shinar). This doesn't contradict Avraham's being Semitic as:
a) His father was an immigrant from "Semite-land" - possibly Syria/Aram/Charan (see Ramban who proves this from Avraham being Semitic and not a Hamitic Chaldean - like the questioner does)
b) There were always descendants of Shem living in Ur- possibly who had never left there (see 10:11).
Some more information on the naming of Ur, and the ancestry of the Kasdim:
I had a look at R' Aryeh Kaplan z"l's Living Torah Chumash (a good source for this kind of thing) and brings the following sources:
1) In a comment on the first mention of Ur Kasdim, Redak writes:
באור כשדים: שהיום נקרא אור כשדים, כי באותו הזמן לא נולד כשד שיקראו בניו כשדים.
That is: The Kasdim/Chaldeans were descended from Avraham's nephew Kesed, and the city was not yet called Ur Kasdim as he wasn't born yet.
2) Josephus and the the apocryphal Book of Jubilees associate Kasdim with Arphachshad - son of Shem.
Jubilees (11:3) writes that Ur son of Kesed (presumably Arpachshad) founded the city around three hundred years before the birth of Avraham. Though this source carries little weight, from both Jewish and academic perspectives, the approach would allow for a Semitic Ur at the time of Avraham.
Rabbi Uri Sherqi suggests that Terah was originally from כנען, which would make it likely that Abraham was born there and/or raised with a Canaanite ethnicity. Why else would they specifically seek to go there?
Alternatively, Abraham is a Shemite as is written explicitly in the Torah. This question could benefit from a precise definition for “ethnicity”.
Scholar feel that the term Chaldees is anachronistic because the Chaldeans did not exist in the days of the patriarchs. It is possible that the term Chaldeans means "Abraham came from where the modern Chaldeans now live" or, it is possible that there was an early group of people who called themselves Chaldeans. Whatever the case, the Bible says that Abraham came from Ur of the Chaldees.
Regarding your second question, Shem was still alive during much of Abraham's lifetime. In fact, some commentaries feel that Shem was Melchizedek. In any case, Abraham was a descendant of Shem, he was a semite. But Abraham was not the first Jew because Judaism did not yet exist.