Here is a response to this part of the question:
When the names of the tribes were written, for example Exodus 28:21,
how would the list appear? Would Joseph be written in place of
Manasseh and Ephraim or would Levi be omitted?
Here is Exodus 28:21 (NRSV):
There shall be twelve stones with names corresponding to the names of
the sons of Israel; they shall be like signets, each engraved with its
name, for the twelve tribes.
There is no definitive answer as to which exact list of names was inscribed on the stones, and in which order. The Hebrew Bible provides no such list in connection with the stones of the breastplate, and the earliest surviving extra-biblical Jewish writings and commentaries come from a period long after the breastplate was lost. So we have no eyewitness accounts of the names and their order on the breastplate.
However, the dominant tradition among Jewish scholars seems to be that the names inscribed on the stones were those of the original twelve sons of Jacob.
Here is one image from Wikimedia Commons that is also reproduced on various Jewish websites:
The names and order as they appear on this graphic are:
Reuben, Simeon, Levi, Judah, Issachar, Zebulun, Joseph, Benjamin, Dan, Naphtali, Gad, Asher
This corresponds to the order in which they are listed in Genesis 35:23-26, which puts the six sons of Jacob's wife Leah first, then the two sons of Jacob's wife Rachel, then the two sons of Rachel's maid Bilhah, then the two sons of Leah's maid Zilpah. This listing occurs in the Bible text right after the birth of Jacob's youngest son, Benjamin, making it the first compact listing in the Bible of the twelve sons, or tribes, of Israel.
The actual birth order (see Genesis 29:31-30:24; 35:16-18) is:
Reuben, Simeon, Levi, Judah, Dan, Naphtali, Gad, Asher, Issachar, Zebulun, Joseph, Benjamin
According to this section of an article from the 1906 Jewish Encyclopedia there are various opinions on the order in which the names were inscribed on the stones. One opinion from an early author is that "the order corresponded to that given in Ex. i. 2-4, except that Zebulun was followed by Dan, not by Benjamin, and that the last two names were Joseph and Benjamin." This rearrangement of the order given in Exodus 1:2-4 (which lists the 11 brothers that went down to Egypt; Joseph, as noted in Exodus 1:5, was already there) would yield this order:
Reuben, Simeon, Levi, Judah, Issachar, Zebulun, Dan, Naphtali, Gad, Asher, Joseph, Benjamin
In this order, the six sons of Leah come first, then the two sons of Bilhah, then the two sons of Zilpah, then the two sons of Rachel, so that the sons of Jacobs two wives bracket the sons of their two maids.
See the above-linked article segment for other opinions.
A fascinating answer to a similar question on the Judaism StackExchange states:
Jacob had twelve sons, and on spiritual matters, we count those
twelve. With Levi as one and Joseph as one. (That's for instance what
you'd find on the High Priest's decision breastplate.)
On financial/land matters, however, Levi did his own thing, and Joseph
got a double portion as his sons Ephraim and Menashe. . . .
So: on spiritual matters: yes Levi, and Joseph=1, total =12. On
monetary matters: no Levi, and Joseph=2, total=12.
The answer does not give a source of this distinction, but it does seem to hold water based on the narrative of the Hebrew Bible and Jewish traditions and scholarship.