If Eve is the "mother of all living" then what of Melchizedek, who is described as "without mother"?

A point to begin is that "living" in Hebrew carries the meaning of fleshly life. It is also important to recognize that in the Epistle to the Hebrews, Melchizedek is presented typologically. When he is described as being "without mother," it refers to the fact that his genealogy is intentionally unrecorded in Scripture, setting him apart as a unique prefiguration of Christ, rather than placing him in the standard genealogical categories delineated in Genesis 3:20.

Problem: Hebrews tells us that Melchizedek, “having neither beginning of days nor end of life, but made like the Son of God, remains a priest continually.” Since Jesus assumed this priesthood (7:21), some reincarnationalists use this verse to prove that Jesus is a reincarnation of Melchizedek. Are they correct?

Solution: No, this is a misuse of this passage. This is clear for several reasons. First of all, it says Melchizedek was only “made like” Jesus, not that Jesus was Melchizedek (Heb. 7:3). Second, Christ was only a priest “according to the order of” (Heb. 7:21) Melchizedek. It does not affirm that He was Melchizedek. Finally, the fact that Melchizedek had a mysterious and unrecorded birth and death (Heb. 7:3) does not prove reincarnation—it was merely used as an analogy for the eternal Messiah, Jesus Christ.