I'm pretty sure you can write what ever you want, considering its a song. In that case, ya. =P
"Some scientists claim that hydrogen, because it is so plentiful, is the basic building block of the universe. I dispute that. I say there is more stupidity than hydrogen, and that is the basic building block of the universe." - Frank Zappa
You can say anything you want in songs.
Try to be discreet about it, it will sound better that way.
What kind of questions? I'd say sure go for it. I mean don't expect an answer but yeah!

Sweet Child o Mine - "Where do we go now?" is the first example that comes to mind.

You can ask open questions like the one in the example above. Or alternatively you can work backing vocals in to have a response effect. I.e. singer asks a question, backing vocals responds. Can add a cool dynamic to your songs as well and encourages the audience to sing that part back to you.

Finally, you can also have backing vocals asking questions underneath the main lyrics. For example, The Living End have bvs singing "How do we know?" again and again in the background during the chorus of the song 'How Do We Know'.

Rhetorical questions are always useful:

"Can this really be true?"
Guthrie Govan = Guitar Genius
It really depends how you do it. I, personally, dislike questions period. That doesn't mean you can't do it if you do it properly. It's hard to do without it sounding terrible, though.

Rhetorical questions are bad, bad, bad. They resolve nothing, it is a loose end that is just hanging there. Especially when they are unresolved. If you have some sort of dialogue-type-thing in your song/poem, then questions are okay as long as you answer them.
Today I feel electric grey
I hope tomorrow, neon black
I'm with Ganoosh that mostly questions sound tacky (unfortunately I posted a whole piece with basically just questions -_-). I do think that closing a piece with a question can be effective, especially if it causes the reader/listener to reflect more on what you said or it provides something that a closed ending couldn't achieve.

I'd limit the extent to which you do it though, but I suppose it can't harm at this stage to experiment.
O! music: Click (Youtube)

^ Click to see an acoustic arrangement of Ke$ha's 'Your Love is my Drug' - everyone's favourite song.
I know I'm a few days late, but I have another good example of questions in a song

"Divide" Disturbed - So can you tell me what exactly does freedom mean,
If I'm not free to be as twisted as I wanna be?
do what u want. Its your song. Thtats the greatest thing about song writing. And that is something ull never get taking away is ur thoughts and ideas. There yours no one else. No one can tell u what is right and wrong. Good luck
My newest cover Rivers Of Babylon sublime style.


My gear:
taylor 310
Fender strat MiM
Cry Baby-GCB-95
Tone port ux2
tascam dp4
80s rock, classic rock, classic metal