There are 26 lines, and based on the title of the puzzle, it seems obvious that they each correspond to a letter of the alphabet.
When I saw the phrase "crushing squashes," I immediately thought of
the band "Smashing Pumpkins."
Then I noticed that that line corresponded with the letter "S" in the alphabet, and it all came clear.
So the puzzle is that
Each line corresponds to a band or musical group of some sort, one starting with each letter of the alphabet.
The list (thanks to those who helped fill it in!):
A) My first is first and second curved. The third is fourth and fourth is third.
Words describe the letters: A = first in alphabet, C = curved, D = fourth in alphabet, C = third in alphabet
B) My first is all; second seventh. All together holy night.
Not sure about the "first". The "second seventh" is a reference to the Sabbath being the seventh day of the week. Not entirely sure about the "holy night," either.
C) First half sickness; second fun. Winter sports, to make a pun.
A cold is a sickness, play is fun. "Cold play" is a punny way of describing winter sports.
D) First half foolish; second hood. All together silly noob.
Daft = foolish; a hood is a punk (think Robin Hood). A "daft punk" could be described as a silly noob.
E) One name singer spitting candy hailing from a state so handy.
He's a singer with a single name, which sounds like the popular candy "M&Ms". He's from Michigan, which apparently resembles the shape of a hand. (thanks, @Hellion)
F) UFOs say au revoir to metasyntactic warrior.
The band name comes from an army nickname for UFOs. Not sure about the "au revoir" or the "metasyntactic", but "warrior" is a synonym for "fighter."
G) Angel thanks for burial. Charity can ease the soul.
The first sentence is a literal interpretation of the band name. The second sentence is a reference to the origin of the band's name, which came from an encyclopedia entry for "grateful dead," which read, "the soul of a dead person, or his angel, showing gratitude to someone who, as an act of charity, arranged their burial."
H) Screeching owl and his friend: swollen dangerous marine.
Hootie and the Blowfish. (Thanks, @BaileyM)
"Screeching owl" refers to "Hootie," and the blowfish is a "swollen dangerous marine" animal. (Although I would argue that "marine" as a noun refers to a soldier, not a fish.)
I) Female torture closes tight. Symbol of medieval might.
An iron maiden is a medieval torture device which is essentially an iron coffin with spikes on the inside that would be slowly tightened around the torture victim. "Female" refers to the "maiden."
J) Classic quintet broken up. Favorite son gets hate or love.
Jackson Five. (thanks, @corsiKa)
They were a classic quintet who broke up. Michael Jackson is loved by some and hated by others.
K) Acronym appearance but came from Peter's former drums
"KISS" looks like an acronym and is often interpreted as "Knights in Satan's Service" or something equally inane, but actually is a play on drummer Peter Criss's former band's name, "Lips."
L) Sometimes vowel; all the same. Distortion of a teacher's name.
"Y" is the "sometimes vowel"; all the vowels are the same in this name. The band is named after the band members' physical education teacher, Leonard Skinnerd.
M) Quite the mix within the troupe. Diacritic misspelled group.
A "motley crew" is a group with a lot of variety. The dots over the "o" and "u" are diacritics, which are being used incorrectly, and the word "crue" is misspelled.
N) Moonlit genies oft appear. Sun is down; desires clear.
When wishing at night, your genie would be lit by the moon. The second sentence is another reference to wishing at night.
O) Above beyond a murder spree. Members change so frequently.
"Over" = "above" or "beyond". "Kill" = a murder spree. This band has had an exorbitant amount of turnover in personnel.
P) Red as bricks with rosy hue. Named for those with color blue.
"Red as bricks" might refer to their famous song, "The Wall." "Rosy hue" is a reference to "Pink", and the band is named after two blues musicians ("color blue"): Pink Anderson and Floyd Council.
Q) Ruler of the whole country. Father was a poison king.
Obviously, the queen is the ruler of the whole country. I'm not sure what the "poison king" refers to. That title seems to refer to Mithridates VI of Pontus, but none of his children were queens of any note.
R) Anger suggested, quite extreme. Robots? No, bureaucracy.
Rage Against the Machine.
Extreme anger = Rage. "The machine" might sound like it's referring to robots, but it is in fact referring to bureaucracy.
S) Crushing squashes makes a mess. Joke name stuck despite protests.
The Smashing Pumpkins.
"Crushing squashes" = "smashing pumpkins", although the "smashing" is actually intended as an adjective, not a verb. The name was originally made up as a joke, but ended up sticking.
T) Five allure but evil not. Classic group has never stopped.
The Temptations. (thanks, @ChrisHayes)
Allure = Temptation, but not evil. This group has been around since the 60s.
U) Name so short; it's not a lot. Named for spy plane? Maybe not.
The band name is very short. There is a U2 spy plane, but its name is not related to the band's name.
V) Try to stop taxi Sienna. Yelling to stop and assist ya.
A Toyota Sienna is a model of van. If it's being used as a taxi, and you're trying to flag it down, you're "van hailin'". (Groan!)
W) Reaching far and wide endemic. Creating world fear pandemic.
Pretty self-explanatory. A widespread panic is a pandemic of fear across a wide area.
X) I heard you like to puzzle so, here's a puzzle in a puzzle, yo.
Xzibit. (thanks, @corsiKa)
This is apparently a reference to the "yo dawg" meme. (thanks, @BaileyM)
Y) Particles from England rock. Temporary name got stuck.
Not sure about the "particles". The group is from England, and had to pick a temporary name for themselves when they found out another act was using the same name. They picked "Yes", and it stuck.
Z) Double letter up above but at the end - sometimes - of stove.
The double letter is Z. "Top" is sometimes at the end of "stove": "stovetop".