I'm not big on apologies anyway.  I'd just chalk it up to "teenage".  I think you might be understandably hyper-sensitive from your experiences with mom and sister.  To be honest, if I asked somebody in my house to turn down the tv and they refused, I'd feel like slamming my door, too.

Next time she comes over, I'd let it pass and save my energies for bigger issues.  I'd also start asking for her specific help more often.  

Hang in there, Clove.  She is a good person, doing well in school, and navigating a tricky home situation.  She is worth it.

I would either just leave it as JRI suggested or approach the topic when she his next over but with some consideration for the fact that you also were involved in the escalation of the argument too by bringing up topics other than her huffing and slamming the door. 

She was trying to do her school work, even if it was not at the time that you felt she should do it. It sounds like she is struggling right now for some reason to complete her school work and that it is out of character for her. She may be feeling fustrated with herself for not being on top of things as much as she usually is and so you pointing that out just played on her insecurities and intensified her fustrations.

I think it's legitimate request to turn tv down. I cant stand loud tv.

I don't think it was a good idea to bring up how she doesn't do chores when she was already upset. Sure she acted out of proportion but you kind of help to escalate the conflict by adding more to the fire. 

online schooling isn't particularly easy and one or a few missing assignments isn't a big deal. It's commendable she wanted to make assignments up. 

id let apology go especially since you contributed to the level of intensity of an argument 

also teenagers are dramatic. It's a given 
 

I am pretty sure Munchkin and my DD are close to the same age... around 13 right? 

My daughter is on this rollercoaster with emotions. Last night she went from crying that she hates her life, hates her family to curled up with me like she was 3 again. Sometimes she calms down, apologizes for being ugly and other times she festers about it for days. It is the age. 

I would let this one go. You are right and recognize that the fight was silly and not really about anything important. Maybe you can even pick up a inexpensive noise canceling earbuds for her so she can plug into her phone and listen to quiet music and not hear the rest of the house? I am not saying that she should be rewarded for being nasty buuuttt this might be one of those solve both your problems things. 

So, I think you and I seem to have similar relationships with our SDs. If I said no to turning down the tv...100% what you described would have went down.

Monday I was helping her with music homework. It's all online but, the last few questions are multiple choice. (You watch a video and throughout the video you answer questions) 
 

Anyways, teacher was teaching about composers and stuff. So she plays a song.. clearly SD or myself haven't  heard it before. Question pops up... "have you heard this song before" a)no b)yes. I was like haha no we haven't it. SD pick no. Thing pops up that it was "wrong" I laughed and was like must be a trick question. 

Now, this is the beginning of the meltdown.... yes  get watery... she's making that OMMMGGGGG face. I NEED TO GET THEM RIGHHHHTTTTT. I was like SD CHILLLLLLL. 
 

Next question, thing pops wrong. I'm like wtf I was pretty sure it was right. Now we have legit tears. I'm like SD I'm walking away now not dealing with the drama. She just starts crying. Whatever.

20 minutes later.... oh I emailed the teacher. (She emailed SD back quick) and she "messed up" on some the questions. Me: Oooooohhhhhhhhh okkkkk. 
 

She didn't apologize for being dramatic. She's 13, some things we have to let go. 

Oh I feel you with those teenage girl vibes!! It's hard to understand and not take it personally, especially when its not your kid, but let me tell you this is normal...I know crazy right?!?!
I got home from work the other day and was trying to talk to BD14 about school and she was telling me she has a test in geography (which she's not doing to hot in). I said well maybe this weekend we can go over the study guide and help you get ready for the test. Suddenly BD14 got defensive, angry, tears are running down her face. I'm so mean for trying to tell her she needs help with school, she basically tells me she doesn't need any help and every time she asks for help I seem to struggle with helping her...oh yes she was on a roll. I told her to stop, calm down and all I was saying is IF she needed help we could help her. But nope too late the drama train was at full steam. She huffed off to her room to cry and slam the door. Yeah so these girls are like little ticking time bombs, and you never know what is going to set them off.

If a teenager huffs off and slams their door, leave them be. Talk about it all later when they are calm.

a bit of PMS teenager happening. I would give her a hug when you see her next and tell her you are sorry the two of you had a fight and you are going to start asking and relying on her more for some help around the house.

Don't convict her based on her sister and mother. She has a lot of your good influence running through her too.

Maybe show a little grace with her and try to remember what it's like with hormones not to mention toxic mom and sis.

You are right more than wrong but sometimes they just need a soft place to fall. You are doing good hang in there .

Back in the day, my counselor told me that if kids can't act out at home (slamming doors, mouthing off), they do it outside the home.

Your spirit is in the right place.  

Once again you have an episode with Munchkin that is unhealthy and you come here to ask if you should disengage. Once again, most say yes and most say your DH is not pulling his weight. But it continues. Isn't that the definition of insanity - trying the same thing over and over again with no different result? I'm not trying to be hard on you but I put myself in Munchkin's shoes and I feel for her. Your relationship with her is getting more toxic and you really need to assess your role in that and make changes.

I am firmly team nurture over nature.  I am also firmly of the perspective that shallow and polluted gene pools are extremely challenging to overcome even when those providing nurture are the full time or near full time parenting influence on the child.

When the source of the shallow and polluted half of a kid's gene pool is the CP, there is nearly zero chance for nurture to prevail over nature.

In the Munchkin's case, she has 2x the influence of the shallow and polluted half of her gene pool.  The direct shit gene source of Toxic Troll and the duplicate toxic influence of the shit Toxic Troll genes carried by the Feral Forger.

Munchkin is extremely fortunate to have both her dad and you in her life.  She knows it, she also knows that her mother and sister are shit.  The shit side of the genetic force is strong and Munch will have a life long struggle against it.  Having her nose firmly rubbed repeatedly in the behavioral crap she chooses to perpetrate is really the only option that you and DH have with her.  The scream/cry and rude crap she recently pulled should be met with new locks installed when she arrives for her next visit.  She has earned the elimination of access to your home without a very humiliating apology to you. DH did a great job of getting his point across and she really had little choice but to apologize to her daddy.  That she has avoided an apology to you escalates the consequences for her chosen behavior towards you.

When she knocks to come in on her next visit, since she will have no key to the new locks, answer the door, inform her that her behavior was unacceptable and she will not enter YOUR home until she apologizes with a very specific

. Then stand there until she does with THE look on your face.  If she starts to build up to a shit gene induced scream/cry shut her down and give the very clear message that how she moves forward from that moment will define the rest of her childhood and potentially the rest of her life because if she chooses to behave as her POS mother and POS sister behave that the love and support she gets from you and her father will take a very unpleasant form going forward with total confrontation of any behavioral shit she pulls.  

You and DH may just get enough time with the Munchkin to counter the shit genes she is cursed with by the Toxic Troll and influenced by Feral Forger.  It will be a closely run thing that will require you and her father to be on your game constantly.  Be very clear that the scream/cry is pure Feral Forger and she is choosing to act as her POS sister acts.

Be wary, be assertive with Munch, and take care of yourself.  You and DH should consider a very relaxing quarantine vacation to a remote though luxurious destination for just the two of you time.  You will both need your bond to be very strong for the run to the finish line in raising Munch.  You may consider bringing Munch out for the last couple of days of your luxury quarantine retreat so leverage that investment to influence her a bit as a transition from the recent highly tense situation to a positive forward focused position.

I am sorry she is baring her shit genes toward  you.

Take care of you.