Upvote:1
The Church of Jesus Christ of Latter-day Saints does teach that angels have free will, including Lucifer and those that followed him, though angelology in LDS teachings is distinct in identifying angels as either future or former inhabitants of this earth.
That shift in perspective differs a little bit from the traditional assumptions offered by the OP though. From the LDS perspective, there is not a separate class of beings to consider and discuss--angels and mankind are the same beings, just in different phases of existence. The free will of Lucifer would be the same free will of Michael, would be the same free will of the rest of the angels, would be the same free will of mortal humankind.
LDS non-biblical canon adds additional commentary on the idea of a third of the angels following Lucifer, but that lies outside the scope of the OP.
Upvote:2
The Catholic Church says yes, angels have free will. See for example paragraph 309 of the Catechism of the Catholic Church:
As purely spiritual creatures angels have intelligence and will: they are personal and immortal creatures, surpassing in perfection all visible creatures, as the splendor of their glory bears witness.
Paragraphs 391-392 of the CCC make it clear that the fallen angels made their own choice, freely:
Behind the disobedient choice of our first parents lurks a seductive voice, opposed to God, which makes them fall into death out of envy. Scripture and the Church's Tradition see in this being a fallen angel, called "Satan" or the "devil". The Church teaches that Satan was at first a good angel, made by God: "The devil and the other demons were indeed created naturally good by God, but they became evil by their own doing."
Scripture speaks of a sin of these angels. This "fall" consists in the free choice of these created spirits, who radically and irrevocably rejected God and his reign. We find a reflection of that rebellion in the tempter's words to our first parents: "You will be like God." The devil "has sinned from the beginning"; he is "a liar and the father of lies".
Your question implies that you believe Michael, Gabriel, and Lucifer are each the "leaders" of 1/3 of the angels. That's not Catholic teaching. The theologians have speculated quite a lot about angels based on the clues in scripture, and I won't present it all here, but here are a few bullet points that may help you compare your theory to the Catholic understanding:
It is believed that there are nine "choirs" of angels ranked from the Seraphim (the highest), to the Angels (guardian angels like yours and mine).
Three angels are named in the Bible: Michael, Gabriel, and Rafael. (Rafael is named in the book of Tobit, which many Protestants don't read. It's believed he is the angel in John 5:1-4 but he is not named there.)
The fallen angels made their choice irrevocably at the moment of their creation, and cannot "change their minds". Likewise the good angels are not in any danger of turning to sin at some point in the future. Their choices are "locked in" like ours will be after we die.
There are many articles about Catholic teachings on angels, such as this one, which you can find by a web search.