I reckon that if you use " NOT" you must use I have not got a car or I haven't got a car you can choose between don't have or have not got, but you can't mix it up. I'm not 100% sure

The main verb comes after the auxiliary verb and the negative, basically. That's the grammar rule, and the meaning is perfectly clear. If you start with "I have not..." then you need to end in a present perfect pattern. We're still waiting for the main verb! If the sentence is about possession, then "have" is the main verb, and you need another auxiliary: "I do not have..."

"I have not a car" is not proper english it is correct to say " I do not have a car" or "I don't have a car" the grammar rule is spelling correctly and using proper clauses, phrases and words in the correct order. it basically makes sure that everything is where it is suppose to be and that it sounds correct when being used in a sentence

The grammar has to do with positive, positive emphatic, and negative declarative sentences. I have a car --- positive I do have a car -- positive emphatic I do not have a car -- negative I go to school -- positive I do go to school -- positive emphatic I do not go to school -- negative

In modern English, you must say "do not V" to negate a verb V. E.g., "I eat meat." --> "I do not eat meat." For the verb "to have," it is possible to say "I have no + noun" instead of "I do not have a + noun." So you can say "I have no car" instead of "I do not have a car."