The Idea in Brief
In both passages (in Deuteronomy and Romans) the "Word of God" is what saves man. According to the Christian New Testament, this same "Word of God" was in direct reference to Jesus of Nazareth.
In the Christian New Testament in Romans Chapter 10, the Apostle Paul identifies Jesus of Nazareth within the Torah, where "the commandment" (הַמִּצְוָה) appears translated as "the word" (הַדָּבָר) in Deut 31:14. That is, this "word" in the Greek Septuagint is ῥῆμα, which is the same word used by Paul in Romans Chapter 10. In both the Torah and Christian New Testament, the "word" descends from heaven and saves man.
By his own statement on various occasions, Jesus of Nazareth had referred to himself as "the son of man," but he also indicated that he had descended from heaven as well (John 3:13). That is, this descending not only included his "incarnation" but also included his death, whereupon this descent continued into "the belly of the earth" (Matthew 12:40). Thus descent carries the wider idea of not only descending to earth for his physical birth but also his death, at which time he continued the descent into "the belly of the earth." The subsequent ascent therefore is his rising; that is, the ascent is his resurrection from the dead. In this regard he returned to heaven from whence he had come (John 13:3).
Going back to the Book of Deuteronomy, the "Word of God" is what saves man. The "Word of God" had descended from heaven, but one need not seek "beyond the sea" to find this word.
Deut 30:11-14 (NASB)
11 “For this commandment which I command you today is not too difficult for you, nor is it out of reach. 12 It is not in heaven, that you should say, ‘Who will go up to heaven for us to get it for us and make us hear it, that we may observe it?’ 13 Nor is it beyond the sea, that you should say, ‘Who will cross the sea for us to get it for us and make us hear it, that we may observe it?’ 14 But the word is very near you, in your mouth and in your heart, that you may observe it.
"The word" highlighted in bold, above, is the same word ῥῆμα in the Septuagint that Paul had used in Romans 10. That is, the "Word of God" saves man. This "Word of God" is Jesus of Nazareth.
Of particular note in the Book of Deuteronomy, the reference to the sea appears to have carried an additional allusion to the underworld. For example, Jonah had entered the underworld through the sea.
Jonah 2:1-3 (NASB)
1 Then Jonah prayed to the Lord his God from the stomach of the fish, 2 and he said,
“I called out of my distress to the Lord,
And He answered me.
I cried for help from the depth of Sheol;
You heard my voice.
3 “For You had cast me into the deep,
Into the heart of the seas,
And the current engulfed me.
All Your breakers and billows passed over me.
That which is beyond the sea therefore may include the idea of Sheol in the Hebrew Bible. That is, according to the account of Jonah, Sheol is beneath the sea "at the roots of the mountains" (Jonah 2:6). Since mountains are made of earth (and not water), their roots would be beneath the seas. In this regard, the idea in Deuteronomy would not only include the idea of "crossing the sea" on the surface of the water in some horizontal sense (like sailing on the surface of the sea going somewhere), but also "crossing the sea" in some vertical sense (as in going downward through the depths of the sea). That is, the Hebrew phrase אֶל־עֵבֶר occurs eight times in the Masoretic Text, and appears to include the idea of "going to the other side" of something.
In summary, this discussion, above, has up to this point attempted to answer the first question of the OP:
What does the passage in Deuteronomy have to do with the discussion in Romans 10?
Now the second question remains to be answered:
Does understanding that connection help clarify how "to ascend into heaven" means
"to bring Christ down"?
The short answer is "yes."
In modern contemporary English, the refrain goes as follows: "What goes up must come down." The apparent understanding of the Apostle Paul was the opposite: "What comes down (from heaven) must go up (back to heaven)." The allusion of the descent of the Lord to earth appears again in the following passage written by the Apostle Paul:
Ephesians 4:7-10 (NASB)
7 But to each one of us grace was given according to the measure of Christ’s gift. 8 Therefore it says,
“When He ascended on high,
He led captive a host of captives,
And He gave gifts to men.”
9 (Now this expression, “He ascended,” what does it mean except that He also had descended into the lower parts of the earth? 10 He who descended is Himself also He who ascended far above all the heavens, so that He might fill all things.)
The citation by the Apostle Paul from the Hebrew Bible comes from Psalm 68, where the context appears to imply that the Lord (YHWH) had descended to Sinai (v.17) because the very same context makes explicit reference to his subsequent ascension on high (v.18). The allusion to Sinai therefore appears parallel to Deuteronomy 30, where the Lord had descended to Sinai in order to give his words to Moses, which will save man.
The Apostle Paul draws the connection between "the Word of God" in the Torah and the "Word of God" in the Christian New Testament. In each case, the hearer need not first ascend into heaven in order to receive revelation, nor need one descend beyond the seas (downward into the underworld) in order to find this revelation either. That is, "the Word of God" has descended and ascended back into heaven, and thus man has the necessary special revelation today that will save him.