This post does not aim to be as comprehensive as last year's post on Eichah.
The breath of our nostrils, the anointed of Hashem, was taken in their pits; of whom we said: "Under his shadow we shall live among the nations." (4:20)
This verse refers to a king of Judah, Yoshiyahu or Tzidkiyahu, who was captured and perhaps killed by enemy armies.
I wish I could tell you what "breath of our nostrils" means, but I have no idea and have no Mikraot Gedolot handy. Perhaps it refers to breathing quickly, which you might do when you were excited, which you might be when seeing your king. That explanation could be totally wrong though.
The end of the verse, while also confusing, can be more confidently explained. In Sefer Bamidbar, when the spies came back and the people were disheartened by their report, Yehoshua and Kalev tried to encourage them with the following line:
"Only rebel not against Hashem, and do not fear the people of the land; for they are our bread. Their shadow is removed from above them, and Hashem is with us; do not be afraid." (Bamidbar 14:9)
That line contains two memorable turns of phrase. The "bread" reference, of course, means that "we're going to totally eat them up in battle", to use a more contemporary expression. The "shadow" reference refers to protection. You are only in a shadow when standing behind something which is bigger than you, and which implicitly protects you. If the Canaanites' "shadow" has left them, it means that (with God's help) there is no longer anything to protect them from the invading Israelites.
Similarly, the "shadow" in Eichah (it's the same Hebrew word too) refers to military protection.
From the books of Shoftim and Shmuel, we see that the people did not ask for a king until their military situation got so desperate under the Judges that they wanted a permanent leader, always ready to go to war on their behalf. As they say: "Let there be a king over us; that we too may be like all the nations; and that our king may judge us, and go out before us, and fight our battles." (Shmuel Alef 8:19-20)
That line from Shmuel is very similar to our line, "Under his shadow we shall live among the nations." Both lines indicate that the king's primary task is to provide military protection, and that being "like the nations" (if perhaps only in the sense of having a standing army) is an important goal.
About 500 years after the monarchy was first instituted by Shmuel, it was effectively abolished by the capture of the Judean king. The political situation instantly reverted to the basically defenceless state the Jews had been in before the first king was appointed. Very soon afterwards, they were completely conquered and exiled from the land.
That is why the fate of a certain single person merits explicit discussion in Eichah, while the rest of the book is entirely anonymous and general in its descriptions.
The elders have ceased from the gate, the young men from their music.
The joy of our heart is ceased; our dance is turned into mourning.
The crown is fallen from our head; woe to us, for we have sinned.
For this our heart is faint, for these things our eyes are dim:
For the mountain of Zion, which is desolate; foxes walk upon it.
You, Hashem, are enthroned forever; your throne is from generation to generation.
Why shall You forget us forever, and abandon us indefinitely?
Return us to You, Hashem, and we shall return; renew our days as of old. (5:14-21)
After a long list of the physical afflictions the people is undergoing, there is a mention of the desolation of the destroyed Temple, before the request for a renewed relationship with God.
One possible reason the Temple is mentioned here is to positively represent the people's state of mind: While they are undergoing these horrible afflictions, they are primarily upset not by the physical pain, but by the fact that the Temple is destroyed. They were punished for not caring enough about spiritual matters, but the shock of the destruction has made them reevaluate their priorities.
Another possibility is hinted at by the verse that follows: "You, Hashem, are enthroned forever; your throne is from generation to generation." Right before mentioning God's eternal existence, Eichah mentions that the Temple is not in fact eternal.
Before the destruction of the First Temple, one of the major problems was people saying that God would never let the Temple get destroyed, so people in Jerusalem were safe no matter how evil their deeds. (See for example Yirmiyahu 7.) Now, by admitting that the Temple is destroyed, and yet hoping for a relationship with God outside the Temple, reparation is made for the mistaken attitudes that prevailed before the destruction.