At the outset, let me explain how narrow this review of the ethics of writing songs really is.
One, it’s not about the ethics of songwriters. No reason to go there.
Two, it’s not about scoring music in songs, singing songs, or pretense of any kind.
Three, it’s only about songs written lately, however loosely you define that.
Last, it’s about the song lyrics—not the individual words, fragments, or repetitions. Just song lyrics fixed in a physical tandem on paper, disc, drives, palms, or bathroom walls.
A lyric is a word, sung. The plural—lyrics—are words that make up a song. Those who write lyrics are lyricists. The meaning of lyrics can be easily missed, ignored, and is often irrelevant to any assessment of ethical compliance.
You cannot copyright words in your song but you can copyright the song. Therein lies the ethical norm—there seems to be only one, or perhaps two. You can copyright your song if you record it—therein lies the two. Every recorded song has two copyrights: one in the written song itself and one in recording the song.
Now comes the “ethics” part of the ethics of writing songs. To be legally copyrighted, the song must be your original work, created by you, showing some minimal amount of creativity. Samo, samo, the ethical norm. Songwriters must write original songs with minimal creativity. Easy peasy. If you’re writing someone else’s song, not yours, and it’s not creative, then it’s unethical.
Questions?
I am an author and a part-time lawyer with a focus on ethics and professional discipline. I teach creative writing and ethics to law students at Arizona State University. Read my bio.
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I am an author and a part-time lawyer with a focus on ethics and professional discipline. I teach creative writing and ethics to law students at Arizona State University. 
My latest novel is Hide & Be. 
If you have an important story you want told, you can commission me to write it for you. 