Quick answer - yes and no.
Take the first two statements. The "to it" at the end is confusing or awkward. "To" is a preposition and "it" is the object. "To" implies something moving toward the object (it). "To sing along to it" implies singing to the song like a singer sings to an audience. This is not wrong, but is an awkward use. To "sing along with it" would be normal use. The use of "had" in "the song had me singing" implies the song made you sing along. Again, this is not wrong, but is not normal use.
The awkwardness of the statements is due to the not normal usage. If a movie reviewer was trying to pique interest in his review, he might say “the movie had me sitting on the edge of my seat the whole time.” The statement begs an explanation as to why. Your statements beg an explanation as to how or why the song had you singing, the book had you bored, or the weather had you sad. Each could be correct, but are not normal use, and they beg context. Without context, they first appear to be wrong or awkward.
So generally, you should not use “had” the way you are. You can be creative and use “had” the way you do, but it has to be in context – you have to explain how or why, or it will appear to be misused.