Melchizedek "is without mother or father or genealogy, having neither beginning of days nor end of life." (v3). If this is literally true, then it strongly suggests that Melchizedek was Jesus, does it not? If it isn't literally true, then how am I supposed to understand this? Also, in what sense does Melchizedek "continue as a priest forever"?

The next passage in the chapter (Verses 5-8) contrasts Melchizedek's collection of a tithe with the Levitical priests' collection of a tithe. The Levitical priests received tithes from their brothers (equals) who were also descended from Abraham. But Melchizedek receives tithes from the patriarch Abraham as his superior, and blesses him (which proves his superiority-- v7). Verse 8 summarizes the argument of 5-7: it says that the Levitical priests were mortal men, but Melchizedek on the other hand is "one of whom it is testified that he lives". So Melchizedek wasn't a mortal man, right? Yet he was a man (v4).

Does that sound like anyone you know? :-)

Also the whole point of superiority to Abraham reminds me of John 8:53ff.

Also verses 15-16: Jesus is qualified as a priest in the same way Melchizedek is. Not by virtue of his descent, but by the power of an indestructible life. So Melchizedek has an indestructible life in the same way that Jesus does, right? He wasn't prevented by death from continuing his office.

I agree that v3 says that he resembles the Son of God, not explicitly that he is the Son, but I don't think resembling necessarily disqualifies being-- especially in the case of theophanies/Christophanies. Consider Daniel 3:25. The Christophany in which the "fourth man" had the "appearance of a son of the gods."