What is the Jewish understanding of
Exodus 6:3,
Short note, Hebrew is read and written from right to left.
the
Tetragrammaton (i.e. YHWH), the Jewish spelling of G-d, and the word Hasem?
The Torath Mosheh Jewish understanding is the Hebrew word (שם) often translated as "name" in English doesn't mean a "pronunciation" but really means "the expression of a characteristic that someone else receives from the one who named as such." So, for example all of the
Hebrew names in the Tanakh for the Source of creation are expressions of what humanity receives from the Source of creation. (אל שדי) "El Shaddai" carries an expression of what humans receive from Hashem while (אלהים) Elohim has a different one of humans receives, so and so on with each name/title. The one you are referencing for example carries the human experience of complete protection. One of the words associated with this (רחם) of Hashem is (רחם) which is the same word used for a womb and related to how a mother's womb protects a developing life.
Also, Exodus 6:3 doesn't make sense to me considering that Eve uses the Tetragrammaton when refering to God at
Genesis 4:1(link):
Again, be aware that you are reading someone's translation w/o a) the actual language it was expressed in and b) w/o the Oral Torah which explains what the words, expressions, and idioms mean. Some of these can only be expressed in Hebrew, because the letters in the Hebrew language also have meanings behind them that can't really expressed in translation w/o HUGE amounts of commentary on every single letter, word, and sentence. Just on this line alone you are asking amount there are loads of information in Hebrew about it going back more than 3,000 years.
Further to this point. All of (בראשית) "Beresheeth" or as you may call it in English Genesis is a type of "cliff notes" type account given to Mosheh ben-Amram (who you call Moses) by Hashem for Mosheh ben-Amram to transcribe in the
Written Torah. I.e. the account of (בראשית) "Beresheeth" or as you may call it in English Genesis was given from Hashem's perspective not the perspective of those written about in it.
Further, Mosheh ben-Amram was given an
Oral Torah to explain how to properly understand what was supposed to be derived from the account/information that Hashem gave about (בראשית) "Beresheeth" or as you may call it in English Genesis. I.e. the Torath Mosheh Jewish perspective is that there are four elements to any reading of the Torah called PaRDeS and the text was constructed for Jews to get certain points out of the text in order to direct Torath Mosheh Jewish life and such.
And then there's Genesis 13:4 and Genesis 14:22.
Same as mentioned above. See my two explainations above.
Also, why do Jews spell God as G-d and use the word Hashem?
Only Jews who use the English language do this. One of the reasons is that it allows them [the Jews who write G-d] to denote that they have a different concept than the various religions who use the English word god. I.e. it is one small way for a Jew who may have to use the English language as medium to make it clear that as a Jew what they are about is distinct from what else is out there and uses that word.
Hashem (השם), is actually in the Hebrew text of the Torah. It is used as a way to not write out the Source of creation's title/name/etc. that deals with the particular aspect that I mentioned above. The reason's why we do that is also from the Hebrew text of the Written Torah and the Oral Torah.