I understand not wanting to be put to sleep, but what they can do instead is a brief infusion of a medicine that will make you sleep through the (very short) procedure. Ask the surgeon, and you will have a chance to speak briefly with the anesthesia people; I am sure there is flexibility.

No, this is deadly.

Can and "should" are two very different things. I would not recommend attempting a tonsillectomy under local anesthesia. Yes, historically, that has been performed, but the risks of doing so are not worth taking, and I would seriously question the judgment of any ENT physician who would accept such risks.