Who is the speaker and who is "him" in Daniel 11:1 according to the Seventh-day Adventist?

I'll try to answer your question. As a Seventh-day Adventist, I have some inside information. :)

But before we get to the answer, it is important to understand some things about Seventh-day Adventists.

• We have a set of beliefs that we regard as fundamental (click here for our list of 28 Fundamental Beliefs), but even those are not set in stone. They are subject to change. As the preamble to the 28FB states: Seventh-day Adventists accept the Bible as their only creed and hold certain fundamental beliefs to be the teaching of the Holy Scriptures. These beliefs, as set forth here, constitute the church’s understanding and expression of the teaching of Scripture. Revision of these statements may be expected at a General Conference Session when the church is led by the Holy Spirit to a fuller understanding of Bible truth or finds better language in which to express the teachings of God’s Holy Word. The Bible is the "definitive revealer of doctrines" (see FB 1). We take Isaiah 8:20 seriously.
• We hold the Bereans in high esteem. (See Acts 17:11.) We check everything by the law and the testimony. Hence, beliefs are subject to change over time.
• Like most groups, and maybe even more than normal, we are not monolithic. We have no creed but the Bible, so it shouldn't come as a surprise that we don't all agree on everything. There are even those who question some of the FB. But that's OK, for the most part. One of my favorite activities with some very close friends is discussing controversial beliefs. It is not uncommon for 3 of us to get together and have 3 different points of view on any given topic.
• Given all this, you should not trust too much what you read on the internet about us, even if the writer is one of us, including me. Strictly speaking, getting an answer from a Seventh-day Adventist gets you the answer for that specific Seventh-day Adventist.
• The most "official" publication is the 28 Fundamental Beliefs, and Daniel 11:1 is not on the list. Sorry.

Now, to your question: Who is the speaker and who is “him” in Daniel 11:1 according to the Seventh-day Adventist? The speaker is Gabriel and "him" is Darius.

The second most "official" publication is the S.D.A. Bible Commentary, so I'll start there. (However, many Seventh-day Adventists, including myself, do not regard it as authoritative and normative. Remember, the Bible is our only creed.)

The angel is telling Daniel of an event that took place in the first year of Darius. In that year Darius the Mede had been honored of Heaven by a visit of the angel Gabriel “to confirm and to strengthen him” (PK 556). Seventh-day Adventist Bible Commentary, Volume 4, page 127

You see PK 556 in that quote. That's a reference to a book by Ellen G. White, who most of us consider to have been given the Spiritual gift of prophecy. Here's what she wrote:

Daniel’s prayer had been offered “in the first year of Darius” (verse 1), the Median monarch whose general, Cyrus, had wrested from Babylonia the scepter of universal rule. The reign of Darius was honored of God. To him was sent the angel Gabriel, “to confirm and to strengthen him.” Daniel 11:1. Ellen G. White, Prophets and Kings, page 556

Here's another quote from her.

Daniel’s prayer in behalf of his people, as recorded in the ninth chapter, was “in the first year of Darius” the Mede. Darius was favored of heaven; for in the first year of his reign the angel Gabriel “stood up to confirm and to strengthen him.” Ellen G. White, The Review and Herald, March 21, 1907

Now here are more quotes from other sources that are more or less well-respected by Seventh-day Adventists.

In Daniel 5:30, it is said: "In that night was Belshazzar the king of the Chaldeans slain. And Darius the Median took the kingdom, being about threescore and two years old." And in Daniel 11:1, the same angel of the tenth chapter says: "Also I in the first year of Darius the Mede, even I stood to confirm and to strengthen him." Darius reigned two years, when he died, and Cyrus succeeded to the kingdom; and as the angel stood with Darius the Mede; and with Cyrus in his third year, to influence him so that Daniel's prayer could be answered, it is certain that it was by the influence of his holy angel that the Lord stirred up the spirit of Cyrus to let go the captive people of God. Alonzo T. Jones, The Signs of the Times, February 11, 1886, page 91

But in the conquest of Babylon, Cyrus was only the leader of the forces. The kingdom and rule were given to Darius the Mede; for, said Daniel to Belshazzar, on the night when Babylon fell, "Thy kingdom is divided, and given to the Medes and Persians." Then the record proceeds: "In that night was Belshazzar the king of the Chaldeans slain. And Darius the Median took the kingdom." Of him we read in Daniel 11:1, the words of the angel Gabriel to the prophet: "I, in the first year of Darius the Mede, even I, stood to confirm and to strengthen him." Alonzo T. Jones, The Two Republics, page 669

The angel of the Lord had stood by Darius the Mede “to confirm and to strengthen him” (Daniel 11:1) in executing the purpose of God upon Babylon. Jeremiah 51:11-14. And through the faithfulness of Daniel, Darius had been brought to the knowledge of the true God. The Advent Review and Sabbath Herald May 17, 1898, page 316 (I believe this is Alonzo T. Jones also, but can't confirm.)

The Medo-Persian empire was born when the time was ripe for Israel's deliverance from the bondage of Babylon. The first king of the united empire was Darius the Mede. He was a man well advanced in life when he came to the throne; threescore and two years old, the record states. But throughout his reign, Gabriel stood by him "to confirm and to strengthen him." Stephen N. Haskell, The Story of Daniel the Prophet, page 161

The angel, after stating that he stood, in the first year of Darius, to confirm and strengthen him, turns his attention to the future. Uriah Smith, Daniel and the Revelation, 1909 edition, page 247

Daniel 11:1 does not contain a significant point of doctrine to us so you won't find a significant treatment of it. I have to admit that I have spent more time researching the verse for this answer than the rest of my life combined. It's just fairly straightforward to us that Gabriel supported Darius.

Most of the material I found was from A. T. Jones. I haven't read his stuff, but I suspect that it has a lot to do with civil rights. You can follow the links I provided to dig through our online databases.