Resolved questions
When we should use I don't have or I haven't

This is different in American and British English.

In American English we say "I don't have X" 90% of the time and almost never say "I haven't." We do sometimes say "I've got X" or ask "Have you got ... X?" but it is rare and usually only in specific informal expressions and situations. The most common question form is "Do you have X?"

In British English, I think people usually say "I haven't got X" and not "I don't have X." They also say "I haven't any X" but I think it is not as common and might sound a little old-fashioned. The question form is "Have you got X?" and not usually "Do you have X?"

Yup, Grant and 博迈克 pretty much sum it up. To summarise:
- It is always 'haven't' when it is an auxiliary verb.
- In American English, it is almost always 'don't have' when it is a main verb.
- In British English, it may be 'haven't' even as a main verb, especially in higher registers of the language, but this usage is steadily growing out of fashion.