#1 is not about a 'rule'. It is simply that in English we BE (am/is/are/etc), not HAVE (have/has/etc), in debt. We can HAVE a debt/debts or BE a debtor.
#2. We use the auxiliary DO to form the negative of:
the present simple: I go, I don't go
the past simple: I saw, I didn't see
the imperative: be, don't be.
We don't use it for the negative of:
the gerund: I enjoy working, I enjoy not working
the infinitive: to be or not to be