Does Abram giving Melchizedek a tithe have a special meaning? Are the bread and wine given by Melchizedek significant? I have heard Melchizedek is a type of Jesus, which makes sense if they are of the same order.

It is certainly true that Melchizedek is a type of Messiah in light of Hebrews 7. However, I would not make too much of the food items that were offered to Abram. Though some see it as a communion service, it is a bit early to have a communion service in the biblical record. That account cannot be used to teach tithing either since Abram did not tithe from his income but tithed from the spoils of war. He gave 10% to Melchizedek and gave the rest back to the king of Sodom. Furthermore, Melchizedek was a type of Messiah but he was not a pre-incarnate Christ since every priest had to be human (Hebrews 5:1) and the Son was not human until the incarnation. Moreover, theophanies made their appearance and then disappeared after fulfilling their mission. They did not hold earthly offices such as king and priest of Jerusalem, which is what Me1chizedek was. It is best just to see Melchizedek as a type of Messiah and leave it there.

[Speaking of Melchizedek] Who is Melchizedek? He appeared to have no father or mother, and there is no record of any of his ancestors. Was he an angel or was he actually the human king of Salem (Heb. 7:1)?

Melchizedek was simply a human being who happened to be both the king and priest of the city of Jerusalem in the days of Abraham. The point of Hebrews is not that he did not have a father and mother, but only that there was no record of it. Hebrews wants to stress that for the Melchizedekian priesthood, ancestry was not relevant as with the Aaronic priesthood. To be an Aaronic priest, one had to show descent from Aaron. But the Melchizedekian priesthood was by divine appointment only, and, therefore, ancestry was not necessary. That is why the Bible does not give the names of Melchizedek's parents or his genealogy. Hebrews 5:1 clearly states that one of the prerequisites for priesthood is that one had to be human. Therefore, Melchizedek could not have been a pre-incarnate Christ nor could he have been an angel. Another reason he could not have been a pre-incarnate Christ is that Old Testament theophanies came and disappeared once they gave their message and never held permanent office on earth; but Melchizedek did hold permanent office(s) as king and priest. In addition, when the Bible compares Christ to Melchizedek, it says he was made like the Son of God. It does not say Melchizedek was the Son of God, but simply “like” in the sense that Melchizedek was a type of Christ as he was both priest and king. Melchizedek was certainly a type of the Messiah, but he was not the Messiah Himself, nor pre-incarnate Christ, nor an angel, but simply a human being.