Have can be an auxillary verb, but get cannot.
To have {noun} means to possess something. The "something" can be a physical object, a physical or mental state, or an outcome.
To have {infinitive} means to be required to do something. It can mean "being forced to" in the sense of that there will be negative consequences if you don't do it. It does not necessarily mean you were held at gunpoint and coerced into doing something.
To have {past participle} is an expression of a perfect tense (past, present or future depending on whether it's had, have/has, or will have). Have used this way does not mean anything in and of itself, it's an auxillary verb and cannot really be substituted with another verb.
To get {noun} means to acquire something. The "something" can be a physical object, a physical or mental state, or an outcome. I worked today and got $200; Today I got sick. Today I got my bills paid off. It may also be used in place where have can be used when the subject is emphasizing that he/she is bringing resources to an event - i.e. acquiring resources for the event. This is a nice party. I got a case of beer.
To get {infinitive} means to be allowed to do something. I get to go to the park.
To get {past participle} means to do something to become a certain state. I got cleaned up. She was unlucky and got screwed.
So, I hope that clears up any confusion you had about the meanings of have and got.
Now, with the above in mind:
I have got this car.
Gotten is the proper past participle of get and is what should be said here, but the above is common (so common that I can't remember the last time I heard I've gotten this car.)
So I have got(ten) this car means I have acquired this car. It's to get expressed in the present perfect tense. Of course, if the person saying this is talking among a group of friends, he/she telling them basically the equivalent of I have acquired this car for us, even though it would also make sense to say I have this car.
This:
I have got to go now.
is breaking the rules, but not uncommon, at least in my location. I gotta go is a way to write the quickly-spoken form of this (but isn't a real word). Avoid using it in a situation where you are expected to appear/sound educated or in writing.
Consider to have got {infinitive} to be a phrase of it's own, and this means the same as to have {infinitive} but stronger. Also don't get it confused with the past tense of to get {infinitive} above.
And, as this says, in American English, have got to is ONLY used in the simple present tense.
