Therefore the Lord himself will give you a sign:

Behold, the עלמה conceives and births a son,

and his name is called Immanuel.

(Isaiah 7:14)

When Jewish translators translated the Hebrew Scriptures into Greek they used the term παρθένος for עלמה and this word a woman who is a virgin.

So does עלמה refer to a woman who is a virgin? In the story of Rebekah in Genesis 24 we read:

Now the girl (נער - na'ra) was very pleasing in appearance. She was a virgin (בתולה - betulah); no man had known her...” (Genesis 24:16)

Then the girl (נער - na'ra) ran and reported these things to the household of her mother.” (Genesis 24:28)

Behold, I am standing by the spring of water. Let it be that the young woman (עלמה - almah) who comes out to draw water and to whom I say, “Please give me a little water to drink from your jar,”” (Genesis 24:43)

The text describes Rebekah as a בתולה (betulah - Genesis 24:16) and a נער (na'ra, Genesis 24:28) and then עלמה (Genesis 24:43). The first two descriptions are of a young virgin girl and so the last description must also include the same ideas. They are used interchangeably of Rebekah before she became Isaac's wife.

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Isaiah 9:6 (5)
"Of the increase of his rule and of peace there will be no end (אֵֽין־קֵ֗ץ)."
לַם רַבָּה הַמִּשְׂרָ֜ה וּלְשָׁל֣וֹם אֵֽין־קֵ֗ץ עַל־כִּסֵּ֤א דָוִד֙ וְעַל־מַמְלַכְתּ֔וֹ לְהָכִ֤ין אֹתָהּ֙ וּֽלְסַעֲדָ֔הּ
בְּמִשְׁפָּ֖ט וּבִצְדָקָ֑ה מֵעַתָּה֙ וְעַד־עוֹלָ֔ם ק
ִנְאַ֛ת יְהוָ֥ה צְבָא֖וֹת תַּעֲשֶׂה־זֹּֽאת׃
This is not referring to Hezekiah but to a king whose rule will be eternal. Notice how this phrase is used in Ecclesiastes 12:12
"עֲשׂ֨וֹת סְפָרִ֤ים הַרְבֵּה֙ אֵ֣ין קֵ֔ץ"
"The making of many books is endless (אֵ֣ין קֵ֔ץ - not an end)."
See also, Ecclesiastes 4:8,16; Job 22:5. The word with the negative has the idea of without end or limit.
Is born or will be born?
Unlike modern Hebrew, Biblical Hebrew has no tenses only actions - complete or incomplete. The tense can only be inferred from the context.

Bethlehem

But you, O Bethlehem Ephrathah, who are too little to be among the clans of Judah, from you shall come forth for me one who is to be ruler in Israel, whose coming forth is from of old, from ancient days.” (Micah 5:2)

וְאַתָּ֞ה בֵּֽית־לֶ֣חֶם אֶפְרָ֗תָה

Here, Bethlehem and Ephratah are synonymous. This is shown in Ruth 4:11: “... may you [בֹּ֫עַז] achieve wealth in Ephrathah and become famous in Bethlehem.”

Both Genesis 35:19 and Genesis 48:7 show conclusively that Ephratah and Bethlehem are references to the same place: "Ephratah (אֶפְרָ֔תָה) which is Bethlehem" [הִ֖וא בֵּ֥ית לָֽחֶם] and "Ephrath (אֶפְרָ֔ת) which is Bethlehem" (הִ֖וא בֵּ֥ית לָֽחֶם).

Genesis 35:16 suggests that Ephrath was the original name of Bethlehem, “Then they journeyed from Bethel; and when there was still some distance to go to Ephrath, Rachel began to give birth and she suffered severe labor.”

To be an Ephrati was to be from Bethlehem. Compare: “Now David was the son of the Ephrati of Bethlehem in Judah, whose name was Jesse...” (1 Samuel 17:12). Jesse was an Ephrati man, of Bethlehem.
So what we read in Micah is this: "You (אַתָּ֞ה = Masc.) Beth Lechem (Masc.) Ephratah (Fem.)... From you (מִמְּךָ֙ = Masc) for me will go out..."
Bethlehem is being addressed.

צָעִיר֙ לִֽהְיוֹת֙ בְּאַלְפֵ֣י יְהוּדָ֔ה
"...being small (literally: small to be) among the families of Judah..."
It is the Ephrati families of Bethlehem who are insignificant in Judah. David was viewed as small and insignificant (1 Samuel 16:11) from a family of Bethlehem (1 Samuel 16:18). Again, it is clear from the association the referent (מִמְּךָ֙ = you) is the place Bethlehem.

מִמְּךָ֙ לִ֣י יֵצֵ֔א לִֽהְי֥וֹת מוֹשֵׁ֖ל בְּיִשְׂרָאֵ֑ל
"From you will come/go out for me (one) to be ruler in Israel..."
'From you', that is, Bethlehem. '...will emerge, come out/go out (יֵצֵ֔א)'. '... a ruler (מוֹשֵׁ֖ל - king by definition). This king goes out in the name of יהוה and will rule in Israel. The messianic implication is obvious: the Messiah will come from Bethlehem!

וּמוֹצָאֹתָ֥יו מִקֶּ֖דֶם מִימֵ֥י עוֹלָֽם
'...and his origins/appearances (מוֹצָאָה lit. goings out, emergings) are from the past (מִקֶּ֖דֶם lit. from before/east), from ancient times (מִימֵ֥י עוֹלָֽם - days of eternity, ancient days).'

So this ruler had made appearances, done things, gone forth, throughout the history of Israel and the Patriarchs. This one who emerges from Bethlehem is no new kid on the block but one who had been around and made numerous appearances to the people. The language relates to the divine and yet this ruler has appeared and acted on many occasions reaching back into the mists of time and eternity. There is a candidate who fits the bill in Tanakh - Malakh Yahweh, the visible, bodily manifestation of יהוה, the God of Israel!
Eternal or of past times?

The Hebrew word ‘olam can be rendered in a temporal—not eternal—way (see Micah 7:14) but it is used to describe God’s eternal nature (Ps. 90:2). Elsewhere in Micah it means eternal  (Micah 2:9,  4:5, 7). Also, Rashi understood ‘olam in Micah 5:2 as eternal. Arnold Fruchtenbaum said, “As regards His human origin, He is to be born in Bethlehem, but regarding His divine origin, He is said to be ‘from long ago, from the days of eternity.’ The Hebrew words for ‘from long ago, from the days of eternity’ are the strongest Hebrew words ever used for eternity past. They are used of God the Father in Psalm 90:2. What is true of God the Father is also said to be true of this One who is to be born in Bethlehem.” "

And he went and lived in a city called Nazareth, so that what was spoken by the prophets might be fulfilled, that he would be called a Nazarene.

(Matthew 2:23)

What prophecy is Matthew referring to regarding Jesus being a Nazarene?

Matthew is obviously not quoting a prophecy directly, as there is no Old Testament passage with the wording he uses. It may be that Matthew is associating the word Nazarene with the Hebrew word netser (“branch or sprout”). The “Branch” was a common term for the Messiah, such as in Isaiah 11:1: “A shoot will come up from the stump of Jesse; from his roots a Branch will bear fruit.” Hebrew was written with only consonants, and netser would have appeared as NZR—the same main consonants as Nazareth. In fact, in Aramaic, the common language of Jesus’ day, the word for “Nazareth” and the Hebrew word for “branch” sounded very much alike. Matthew’s point could be that Jesus was “sprouting up” from an obscure village in Galilee; Jesus was the Branch predicted by the prophets, and the name of the town He grew up in happens to sound just like the prophets’ word for “branch.”

Another option is that Matthew uses the word Nazarene in reference to a person who is “despised and rejected.” In the first century, Nazareth was a small town about 55 miles north of Jerusalem, and it had a negative reputation among the Jews. Galilee was generally looked down upon by Judeans, and Nazareth of Galilee was especially despised (see John 1:46). If this was Matthew’s emphasis, the prophecies Matthew had in mind could include these two passages concerning the Messiah:

“But I am a worm and not a man, scorned by everyone, despised by the people. All who see me mock me; they hurl insults, shaking their heads” (Psalm 22:6–7). It’s true that Nazarenes were “scorned by everyone,” and so one could see this messianic prophecy as an allusion to Jesus’ hometown of Nazareth.

“He was despised and rejected by mankind, a man of suffering, and familiar with pain. Like one from whom people hide their faces he was despised, and we held him in low esteem” (Isaiah 53:3). Again, in Jesus’ day, Nazarenes were “despised and rejected,” and so Isaiah’s prophecy could be viewed as an indirect reference to Jesus’ background as the supposed son of a carpenter from Nazareth.

If Psalm 22:6–7 and Isaiah 53:3 are the prophecies that Matthew had in mind, then the meaning of “He shall be called a Nazarene” is something akin to “He shall be despised and mocked by His own people.”  His upbringing in an obscure and despised town served as an important part of His mission. Jesus identified Himself as “Jesus of Nazareth” during His encounter with Saul on the road to Damascus (Acts 22:7–8). After his conversion, Paul mentioned Jesus of Nazareth (Acts 26:9). One of the names of the early Christians was “Nazarenes” (Acts 24:5), and the term Nasara, meaning “Nazarene,” is still used today by Muslims to identify a Christian.

Hosea 11:1

“...This was to fulfill what the Lord had spoken by the prophet, “Out of Egypt I called my son.”” (Matthew 2:15)

It is claimed that this refers to Israel and not Jesus.
The rabbis used several modes of Biblical interpretation - p'shat, remez, midrash (drash) and sod. This is probably a remez, a hint of a very deep truth. Israel is called God’s son as far back as Exodus 4:22. The Messiah is called God’s son a few verses earlier, reflecting Tanakh passages such as Isaiah 9:5–6(6–7), Psalm 2:7 and Proverbs 30:4. As Son of God the Messiah is viewed as one with Israel. Using remez, Matthew is using Hosea 11:1 to show that Yeshua is representative of the nation Israel.

David H Stern says in the JNTC "the idea that the Messiah personifies or is identified intimately with Israel is a Jewish one. First of all, we see it in the Tanakh itself. Compare Isaiah 49:3 (“You are my servant Israel, in whom I will be glorified.”) with Isaiah 49:6 (“Is it too slight a thing that you should be my servant … to restore the preserved of Israel?”). The servant is at once Israel and he who restores Israel, that is, the Messiah. In chapter 12 of Raphael Patai’s The Messiah Texts he quotes Pesikta Rabbati 161–162, where the Messiah is called Efrayim (a name symbolizing Israel) and is at the same time presented as bearing Israel’s sufferings. Likewise the thirteenth-century work which is at the core of the Jewish mystical approach called kabbalah, the Zohar (2:212a), links the Messiah’s suffering with that of Israel. Patai also retells the eighteenth-century Rabbi Nachman of Bratslav’s story of the viceroy and the king’s daughter, adding that most interpreters understand the viceroy to represent both Israel and the suffering Messiah."

Psalm 110
Daniel 9
Psalm 22
Jeremiah 31 - The New Covenant
Descendant of David