"Get a car" and, to paraphrase, "ha ha, your mom wouldn't drive you. You're a loser" are the standard, glib contributions from jerks. While you don't give a lot of information, no one seems to stop to consider that you have a car and it's in some form of dysfunction, that you can't afford one yet but are working toward it (hence having a job), or just generally that you do the best you can and they should not be dicks to you.
As to your job: if they're really that strict about tardiness, fuck 'em. Again, I don't know your work habits but, from what you've said, that's unfair. Sadly, there's not much you can do if an employer wants you gone. My best advice, assuming that you would like this job back, would be to continue to GENTLY persuade your mother to speak to someone in authority. If her pride is getting in her way, figure out an approach where she benefits from the situation (ie you get your job back, get paid, and buy a car so she needn't be bothered) or doesn't feel that she's admitting to being unconcerned and lazy (ie have her call and say that you both did the best you could under the circumstances but that she can assure them that, as your back-up ride, you'll have a way there from here on out... whether any of that is true or not).
It sucks that this happened and it sucks that you've gotten the responses that you have.