I don't believe in holding people to perfect standards or anything, but I can't see a single example of healthy communication in this whole thing.
DH threw her clothes at her and shouted at her to get back upstairs? Then shouted at you?? This is ridiculous and totally unnecessary on both counts. Even if DH believes you made a mistake with the washing, shouting at you is completely over the top and I don't understand why he feels he can do that to you.
DD reacted badly to her clothes being ruined, and you shouldn't have to tolerate being shouted at by her, either. Is it customary in a row for DH to start shouting and throwing things (I know it was only clothing so I'm not insinuating he smashes the place up or anything, but still)? If so this might well explain why she defaults to shouting when something goes wrong.
So she's stormed out and now you're in tears, wondering if you've done something wrong. Actually, all that happened was you made a mistake while doing a household task and got shouted at for it by two people, completely unfairly. Yes, ideally you wouldn't have ruined the clothing but it's not like you did it on purpose. I'd be asserting some firm boundaries right now about how I'm treated in the house.
I don't think it was necessary to be so harsh to DH regarding the flat tire. I wouldn't feel comfortable knowing a 19yr old was sat waiting in a car, who knows where, at night, for it to be repaired.
Basically it sounds to me like there's no solid foundation for healthy communication here at all. Why are they so shouty? Why is DH throwing clothes at DH telling her to get upstairs? Seems like there are issues.