The author of Hebrews has taken a psalm that had nothing to do with messianic prophecy and nothing to do with Jesus, but saw Jesus painted all over it, and brought it to bear on his identity and nature. It gives the Son:

1. Equality with God
2. Universal Sovereignty
3. Divine eternity
4. Divine sinlessness (righteousness), holiness and purity.
5. The king of God's covenant
6. God's ambassador on earth: God with us, God incarnate
7. The one who will fight for God's kingdom
8. The one who will accomplish justice on the earth for God

The thought starts in verse 8: "But about the Son he says..." He starts with a quote from Ps. 45.6ff., a psalm that is a Hebrew nuptial ode for a king, treated by the author of Hebrews as Messianic because, he is asserting, in Jesus the David monarchy achieves its fullness of definition. The rabbis did interpret Ps. 45.7 (Heb. 1.9) messianically.

Then Hebrews 9 passes on to Ps. 102, which is not taken by anyone to be Messianic. So how can these words justifiably be applied to the Son? First, he has already said the Son and the Father were one and the same (in v. 2, speaking of creation). Second, since the Son was the intermediate agent in the work of creation as Creator God, he is entitled to the designation "Lord."

As far as Heb. 1.6, the referent is more likely Ps. 97.7 than Dt. 32.43.

I don't regard the language of Heb. 1.10 resting on a misinterpretation, but on reasoning: Since Jesus was the creator, he is therefore God, and therefore eternal, unlike the angels.

What reason is there to believe (as Hebrews 1:10 claims) that Psalm 102:25-27 is God the Father speaking about Christ, and not simply the Psalmist speaking about God?

The crux is that Hebrews 1:10 portrays Psalm 102:25-27 as a statement of God himself to his Son; and yet it had clearly been the human Psalmist speaking (imploring God to spare/prolong his own life) in the verses immediately before this, and there's nothing to indicate that a shift of speakers has taken place.

And just to be clear, I'm interested in answers that aren't just question-begging—so nothing that along the lines that Hebrews simply must be correct here, even if there's no evidence to suggest this (or especially if there's good evidence against this).

Naturally, I don't think that there's any reason to believe this, and think that Hebrews has simply misinterpreted this.

Incidentally, the quotation in Hebrews 1:6—used to suggest the superiority of Christ to the angels—also seems to use an altered text of Deuteronomy 32:43. Hebrews 1:6 says "all God's angels," πάντες ἄγγελοι θεοῦ, whereas the Septuagint of Deuteronomy 32:43 actually reads πάντες υἱοὶ θεοῦ, "all God's sons," with "sons of God" being a well-known polytheistic/henotheistic Semitism for the actual offspring of literal gods (who are gods themselves), and not simply angels as they were later interpreted.

Most importantly, however, the earliest text of Deuteronomy 32:43 found among the Dead Sea Scrolls, in 4QDeutq, simply reads כל אלהים here, "all gods." (Compare also Psalm 97:7.)

This text from the Dead Sea Scrolls likely represents the original text of Deuteronomy 32:43, which was later modified for theological reasons in the Greek versions (and removed from later Hebrew manuscripts, as well) to more clearly suggest non-divine beings / a stricter monotheism. Incidentally, the alteration here is similar to what happened earlier in the Deuteronomy 32, too, with 32:8.

So we'd certainly have precedent for the language of Hebrews 1:10 itself resting on a misinterpretation, whether deliberate or unintentional, of earlier texts from the Hebrew Bible.

(Of course, that's not to say that the author of Hebrews didn't think of Christ as God, independent of these things. I'm simply saying that some of the argumentative support for this that he adduced was based on interpretive errors.)