Explanations/Questions:

This is another arc that for me has had a variety of shapes through several revisions. The smaller units are straightforward enough, the challenge is in how these 'second-tier' arcs then relate to one another:

v.12-13, Don't be surprised at the trial, rather, rejoice

- - - - -

v.14, If you are insulted for the Name, you are blessed

γἀρ (for)

v.15-16, Make sure no one suffers as a criminal, but if as a Christian, glorify God in that Name

ὅτι (for, because)

v.17a, it is time for judgment to begin within God's house

δὲ (and, but)

v.17b-18, if it begins here, those outside in disobedience will not stand

ὥστε (therefore)

v.19, those suffering in accord with God's sovereign design should stake their lives on a faithful Creator.

v.14-16, What relationship does γάρ (v.15) indicate?  There seems to be an atypical role for γάρ here. Most English versions leave it vague in translation. NASB gives it the force of a mild exclamation, "Make sure..." (similar to 1 Cor. 11:22, where it is the forceful, "What!"). See next question...

Is v.14 primarily connected to v.12-13 or v.15-16?  Schreiner (New American Commentary) sees it as a distinct thought added to v.12-13 which together are then Explained (γάρ) in v.15-18. This gives γάρ a strong role in the overall structure of the passage.  My arc however, ties v.14 to v.15-16 under the theme of 'the Name': v.14, ...insulted for the name of Christ; v.16, ...as a Christian; v.16, ...glorify God in that name.

What specifically does v.17a support?  That it is now time for God to be refining his people logically supports a call to joyful alertness (v.12-13), as much as it gives a reason why suffering insult is a sign of blessing (v.14), and probably more than solely supporting unashamed suffering as a Christian rather than as a criminal (v.15-16). 

And is v.17b-18 part of that support or does it add a new point? Though Peter repeats the point from v.17a, he does so in order to introduce an entirely new thought - that those outside the household of God by their disobedience to the gospel are doomed (i.e. What shall come of them?). He reaffirms this point by quoting Proverbs 11:31 (Septuagint).

Is v.19 an inference from v.17-18 or from the whole of 12-18? It is the realities of God's judgment - the refining of his people and the damning of the ungodly - that lead to the imperative to trust in God's sovereign design in suffering and his faithfulness as the author of life. But v.19 also stands as a conclusion to the whole in that his flow of logic has progressed through v.12-18 and culminates with this call to entrust your souls to a faithful Creator.

Summary: Trials should not come as unexpected surprises but as a cause for rejoicing in anticipation of even greater joy  at Christ's appearing.  Moreover, to suffer for the Name is to be blessed and is no cause for shame but an opportunity to honor God in that Name. Indeed, God's judgment begins with his own household. And be assured that that which is aimed to purify his people will be the end of those opposed to the gospel. So as you suffer in accord with God's soveriegn design, stake the whole of your life on the Faithful One who is the Author of life.

Insights:

v.13d, If the participle is not merely providing an additional possibility (...that you may rejoice and be glad) but detailing the manner of the rejoicing (...that you may rejoice being exceedingly glad), then the point of the Ac-Pur argument is that the aim of joyfully expecting suffering in the present is increased and amplified rejoicing at Christ's revelation.

v.17-19, It is the gospel, and the gospel alone which separates mankind. The dividing line is obedience - and what is it to obey the gospel? It is the obedience of faith in which you stake the whole of your life on the faithfulness of the God who has made you.

v.19, "...while doing good."  Peter never leaves righteous living off the table.  Holy conduct does not depend upon the circumstances of our lives.

Central Idea

God will reverse every curse he has made against his people and give them the new core identity of "once cursed due to sin, but now blessed by the will and provision of the Lord in grace."

[[ Note: English versification has been utilized for both arc and notes. ]]

How are the major pieces of this passage connected?

Again the Lord is speaking about "that day" of grace in the future. There are three mainline clauses (weqatal verbs in Hebrew) in this passage that speak directly to the three curses of 1:2-9. (There, as you will remember, God had Hosea marry a whore who bore to him three children who were in turn given names to describe God's judgment and rejection of Israel.)

1. Vs23a: God will sow the nation for himself in the land. This relates to the child Jezreel born in 1:3-5 where he represents coming punishment of God for blood that was shed at the place named Jezreel. Here, however, the meaning is transformed. Instead of being a symbol of judgment, the linguistic meaning of the word Jezreel ("God will sow") finds fulfillment. This verse explains how "the day of Jezreel" in 1:11 looks to a future day of hope.
2. Vs23b: "No Mercy," the second child (1:6-8) is promised mercy from God.
3. Vs23c: Finally, to "Not My People" (1:8-9) God says the sweet words, "You are my people."

What is all the answering about?

First, we must understand a little more about the verb "ענה" ("to answer") in Hosea. There are four uses outside this passage, two of which seem especially helpful in illuminating Hosea's usage. In 2:15, God says that Israel will "answer" as in the days of her youth with the apparent meaning of "willingly respond to instruction and guidance." In 14:8, the meaning is similar and even more pertinent to our passage. It reads: "O Ephraim, what have I to do with idols? It is I who answer and look after you. I am like an evergreen cypress; from me comes your fruit." Here, Israel has been foolishly requesting provision from idols when in fact it is the Lord who "willingly responds to [her] requests" (an amplified translation of "answer" in this verse).

So now to passage. What is going on with all the answering? In a sentence, God is showing himself to be the beginning and end of blessing delivered to his re-established people. Here is how it works:

1. God declares to sow the people in the land, which includes the land producing bountifully. (Notice "Jezreel"--meaning "God will sow"--in vs22 and all of vs23a)
2. God's declaration, therefore, requests--as it were--the grain, wine (grapes) and oil (olives) to grow and flourish.
3. But the grain, wine and oil need good soil to do so. And so they in turn request the earth (soil) to provide nourishment.
4. But the earth needs rain to become good soil and so follows by requesting the heavens to shower down.
5. But it is God that gives rain (Job 5:10) and so the heavens put in their request to God.
   [[Now with that setup, the situation unfolds from vs21c]]
6. God answers the heavens and provides rain (21c).
7. The heavens then answer the earth and send the showers. (21d).
8. Next the earth answers the grain, wine and oil and provides nourishment (22a).
9. Finally, the grain, wine and oil answer the declaration of God with a harvest.

Thus, God puts in the request and God provides the basis of the result. “For from him and through him and to him are all things. To him be glory forever. Amen.” (Rom 11:36) See also Zech 8:12 and Ezek 36:8-9.

What is the new identity of the people?

Notice that in this particular passage, God does not rename the children. Rather, he speaks blessing to them while still using their curse names ("Jezreel," "No Mercy," and "Not My People"). There is something very significant in this. The identity of this restored people of God is not just simply "the chosen people of God," but rather "those who were once cursed by God due to sin but now are graciously received in with unspeakable blessing." This is the New Covenant identity, and something huge that Peter picks up as seen below.

How is this passage connected with Romans 9:25-26 & 1Peter 2:9-10?

With regard to Rom 9:25-26, see the arc of Hosea 1:10-11 from the archives, noting the notes section entitled, "How does Paul's use of this passage in Romans 9:26 work? How does that affect our interpretation?" 

Peter does much the same thing with his paraphrased quote of 2:23. "You" in 1Pet 2:9-10 is the recipients of the letter--"the elect exiles" scattered about--those who obey Messiah Jesus and are sprinkled by his atoning blood, from both the Jews and the Gentiles. And thus, as Paul does in Rom 9:25-26, Peter interprets the book of Hosea to be speaking here of a transformed people. Still Israel, yes, but not still the same. Those who have rejected the Messiah among the Jews are no longer included, while those who have received him among the Gentiles are full members. All who are now a part of this people where beforehand "not a people" and "had not received mercy"--for Gentiles because this was always their state, and for Jews because of their rejection by God as recorded in the book of Hosea due to their rebellion.