Do you ever find yourself singing a song you are making up on the spot (to calm a child for instance) and then realize halfway through that it's a prayer tune?

Do you finish the song?

Feel bad that you're degrading the tune or feel good that you're exposing your child to it?

Does the topic matter? (e.g., sharing toys vs. changing a diaper)

Not sure if I do it -the songs I make up are all spontanious and forgotten right away. It wouldn't bother me though since I don't consider the tune itself to be holy. There is also something to be said for having the kid feel familiar with the tunes but I wouldn't overrate it.

Would it depend on the tune? For instance, a popular but recent Adon Olam tune vs. Kol Nidre.

As if these tunes are mi-Sinai? Most were picked up from the surrounding non-Jewish culture.

Valid point. Just like the tunes for Adon Olam that sound suspiciously like Yankee Doodle and Gilligan's Island.

However, for many people, the tunes are linked in their mind to the particular prayer so singing about a poopy diaper to the tune for Kol Nidre (for example) might feel inappropriate.