Most Helpful Girl
She may have been ignorant and insensitive about how to pick up a dog, and was therefore abusive to him. But you were mean and insensitive to your date and behaved cruelly towards a woman because of your abusive reaction. You don't stop abuse with abuse, especially when the person may not have known the correct behavior. You wouldn't even talk to a misbehaving animal like that! Especially one who was learning something new.
She may have only had experience with seeing cats carry their young by the scruff. And her being concerned enough to bring her dog on a date shows she's not out to harm him. Many rational people may feel just as angry with how you treated her as you claim you were angered by how she treated the dog.
I think you were more angry than necessary because you felt you should've alerted her about how she was lifting him when you first saw her do it, (you felt shame/guilt) because it makes you feel like you are participating in animal abuse. so to make up for it (subconsciously) and prove to yourself that you care about animals (and also scold yourself), you overdid it and ended up abusing the woman in public. Oh, and the dog had to also witness the yelling/cursing as well.
Peaceful, loving people communicate in peaceful, loving ways. You seem to think of yourself as a person that cares about the wellbeing of others, people or animals. If that were true, you would acknowledge that your way of speaking to her was unacceptable.
If I were you, I would wish I handled it better, and would've taken the opportunity to teach her kindly after letting her know that what she was doing was dangerous without spewing curse words and judgements at her. Even if you had done that, it would've been hard for you to get over what you saw in the first place to want to see her again. But now, because you berated her, you not only owe her an apology, but you're obligated to direct her, kindly, to proper dog caring/ training resources and wish her luck.
The last question you posed should be more concerned with how unacceptable you were than how unacceptable she was. Any self respecting woman would promptly decline your invitations to get a second chance from her after how you handled, what was likely, an honest mistake she committed.
Most Helpful Guy
You sound like a total douchewaffle. What seems like common sense to you isn't necessarily common sense to someone else. For instance, I think it would be common sense to help out a new dog owner by kindly pointing out that picking a dog up by the collar is probably not the best thing to do and then show her how to do it. I wouldn't think it would be common to yell at her and call her a fucking cunt, but apparently that is common sense to you.
The good news is you now have a perfect way to determine if you should go out with her again. If you ask her out again and she willingly says yes you should immediately dump her because she is likely a few bricks shy of a full load after the shit you pulled. If she insists on an apology from you then you might have a keeper. Good luck!