Which angels and fallen angels are named in the Bible. and what are they responsible for?
There are only two angels mentioned by name in Scripture:
Michael, the archangel, mentioned in Daniel 10 (and also in Jude). He is described as fighting the prince of Persia.
Gabriel, who announced the birth of Jesus to Mary in Luke 1. Here Gabriel is being an angelos quite literally - a messenger.
Other Angels appear in the OT (One to Manoah in Judges to announce the birth of Samson, three to Abraham in Genesis), but they are unnamed.
An angel has only one job, as implied by the name - they deliver messages.
When they are not doing this job, they are usually in the presence of the Lord, praising him and announcing his glory (Isaiah 6, Revelation)
Debatedly, the only fallen angel is Lucifier in Isaiah 14 - the "Star of Morning" who is the Devil.
One could also describe "Legion" as a demon in Luke 5 (at Genesseret), but that isn't a proper name, rather a collection of Demons.
@Affable Geek's right, but doesn't mention the Archangel Raphael as his tradition doesn't consider the book of Tobit scripture.
Since Catholics and Orthodox include this book, he's also one of the angels named in the Bible. His name means God Heals and he's the one who took Tobias on his journey and helped him heal his father's blindness. He's actually pretty awesome.
Also, in Catholic tradition (and other traditions), it was two angels who visited Abraham and Sarah, along with God.
...the truth of the matter seems to be this, that one of them was the son of God in an human form, that chiefly conversed with Abraham, and who rained from heaven brimstone on Sodom; and the other two were angels in the like form that accompanied him in that expedition:
I have been rereading many texts I thought I knew well, and finding new surprises. Revelation 9:11 (ESV) says: "They have as king over them the angel of the bottomless pit. His name in Hebrew is Abaddon, and in Greek he is called Apollyon." And, we may not know until all things are revealed what the true roles of the angels actually are. But here is another named angel (other than Michael, Gabriel and Lucifer).
The source to the list of fallen presented from A-Z comes from "A Dictionary of Angels including the fallen angels" by Gustav Davidson and is considered one of the preminent sources on angels. The word "demon" comes from The words dæmon or daimôn which are Latinized versions of the Greek "δαίμων", a reference to the daemons of ancient Greek religion and mythology, as well as later Hellenistic religion and philosophy. Daemons are benevolent or benign nature spirits, beings of the same nature as both mortals and gods, similar to ghosts, chthonic heroes, spirit guides, forces of nature or the gods themselves (see Plato's Symposium). Walter Burkert suggests that unlike the Christian use of demon in a strictly malignant sense, “[a] general belief in spirits is not expressed by the term daimon until the 5th century when a doctor asserts that neurotic women and girls can be driven to suicide by imaginary apparitions, ‘evil daimones’. How far this is an expression of widespread popular superstition is not easy to judge… On the basis of Hesiod's myth, however, what did gain currency was for great and powerful figures to be honoured after death as a daimon…”  Daimon is not so much a type of quasi-divine being, according to Burkert, but rather a non-personified “peculiar mode” of their activity.
In Hesiod's Theogony, Phaëton becomes an incorporeal daimon or a divine spirit but, for example, the ills released by Pandora are deadly gods, keres, not daimones. From Hesiod also, the people of the Golden Age were transformed into daimones by the will of Zeus, to benevolently serve mortals as their guardian spirits; “good beings who dispense riches…[nevertheless], they remain invisible, known only by their acts”. The daimon of venerated heroes, were localized by the construction of shrines, so as not to restlessly wander, and were believed to confer protection and good fortune on those offering their respects.
Characterizations of the daemon as a dangerous, if not evil, lesser spirit were developed by Plato and his pupil Xenocrates,[dubious ] and later absorbed in Christian patristic writings along with Neo-Platonic elements.
In the Old Testament, evil spirits appear in the book of Judges and in Kings. In the Greek translation of the Septuagint, made for the Greek-speaking Jews of Alexandria, the Greek ángelos (άγγελος: "messenger") translates the Hebrew word mal'ak, while daimon (or neuter daimonion) carries the meaning of a natural spirit that is less than divine (see supernatural) and translates the Hebrew words for idols, foreign gods, certain beasts, and natural evils. The usage of daimon in the New Testament's original Greek text, caused the Greek word to be applied to the Judeo-Christian concept of an evil spirit by the early 2nd century AD.
Satanists have used the word demon to define a knowledge that has been banned by the Church.
Wikipedia: Daemon (classical mythology)
There are many fallen angels among whom only Lucifer is named.
“And there was war in heaven: Michael and his angels fought against the dragon; and the dragon fought and his angels, And prevailed not; neither was their place found any more in heaven. And the great dragon was cast out, that old serpent, called the Devil, and Satan, which deceiveth the whole world: he was cast out into the earth, and his angels were cast out with him.” (Revelation 12:7–9, KJV 1900)
Enoch 1 talks about the sons of God that mates with the daughters of man being the fallen host in hell. If you follow Revelations, the rebellious angels under Lucifer are still in Heaven awaiting Michael to release them for the war in Heaven to come to earth:
7 Then war broke out in Heaven; Michael and his angels battled against the dragon. The dragon and its angels fought back, 8 but they did not prevail and there was no longer any place for them in Heaven. 9 The huge dragon, the ancient serpent, who is called the Devil and Satan, who deceived the whole world, was thrown down to earth, and its angels were thrown down with it.
10 Then I heard a loud voice in Heaven say: "Now have salvation and power come, and the kingdom of our God and the authority of his Anointed. For the accuser of our brothers is cast out, who accuses them before our God day and night. 11 They conquered him by the blood of the lamb and by the word of their testimony; love for life did not deter them from death. 12 Therefore, rejoice, you Heavens, and you who dwell in them. But woe to you, earth and sea, for the Devil has come down to you in great fury, for he knows he has but a short time. "
Satan means adversary in Hebrew and was either not the original name to the leader of the rebellious whether he be Lucifer or Sammael (who Rabbinic lore is the "Chief of Satans") or Satan is a title for a group of angels that leads the rebellious to fall. Let it be noted that Lucifer is included in Gustav Davidson's list of the fallen as #57 by alphabetical order though he was left out of the list posted a few posts before mine.
There are more then one fallen angel as Lucifer (the morning star) is a angel an he was the reason of the fallen because he wanted his own kindgom so therefore Jehovah banned him and any of the angels that agree with lucifer. He is the ruler of hell and it his job to test both humans and the fallen. Sammeal/Satan is the only archangel to fallen and he rules in hell as the prince. Ori was one Fallen it was his job to hell luicfer by killing any rouge demons. Also he was excellent at seducing both men and women. Gusion (fallen) was known for seeing the future, he was also Ori friend, it was said that he help Ori in battle against Sarteal. Sarteal (fallen) want to to rule hell instead of Lucifer.
Some of the fallen mated with humans and produced nephilim (half-angel half-human) There are seven archangels and they are 1. Uriel, one of the holy angels, who presides over clamer and terror 2. Raphael, one of the holy angels, who presides over the spirts of men 3. Raguel, one of the holy angels, who take vengeance on the world of the luminaries 4. Micheal, one of the holy angels, to wit, he that is set over the best part of mankind and chos 5. Sarakiel, one of the holy angels, who is set over the sports, who sin in the spirt (and the only female archangel) 6. Gabriel, one of the holy angels, who is over paradise and the serpents and the cherubim 7. Remiel, one of the holy Angels, whom god set over those who rise - the book of Enoch 20:1-8 Therefore Micheal can't be a fallen.