"Haven't" as a helping verb (no he/has/ha/hemos/han + participio) is quite natural, but using it as a main verb to express possession (no tengo/tienes/tiene/temos/tienen + complemento directo) would sound extremely formal in American English. I always recommend that students use "don't have" and "doesn't have" to avoid sounding like they've just stepped out of a Shakespearean play.

I agree with both of the other answers: "don't have" being a common negation of "to have", while "haven't" is more commonly a negation of the auxiliary verb. while haven't can also be a negation of the transitive verb ("I haven't any money"), I think it is fair to say that that usage is dead in the United States. I still hear folks from the UK, Ireland and India use it. I do not hear it from Canadians. I do not know about other big English speaking hubs.

There is no difference whatsoever. You can say "I don't have any time" or "I haven't any time" The first is more common, but the second sounds more refined, more stylish. In conversation, people often insert the ubiquitous horrible word "got": "I haven't got any time" Feel free to do this, but "got" adds nothing to the sentence and is not needed.

"Don't have" and "haven't" both indicate a lack or absence of something, but they are used in different contexts. "Don't have" is the contracted form of "do not have" and is typically used in the present tense. For example: "I don't have enough money to buy a new car." "Haven't" is the contracted form of "have not" and is used in the present perfect tense. It indicates an action that started in the past but has relevance to the present. For example: "I haven't visited my grandparents in a long time." This implies that you have not visited them recently, emphasizing the duration of the absence up to the present moment. In summary, "don't have" is used in the present tense, while "haven't" is used in the present perfect tense to indicate an action that started in the past and continues up to the present or has just been completed.