Could I be questioned for a crime even if I'm drunk during the time of inquiry?

Of course you can. There is no law that says drunk people can't be questioned. With that said, you have the right to remain silent. Only idiots talk to investigators/cops. Remember, no matter what you say it will be twisted around and used against you later.

If you are questioned during a period of voluntary intoxication it is doubtful that there was anything illegal simply as a result of your drinking. There are other questions which arise, like were you a target or placed under arrest and, perhaps, whether or not you were due to be read your Miranda rights. You should engage an attorney and provide all the details seeking a firm opinion.