I have 3 kids myself (18, 14, 12) and my husband is their step-father. I know first-hand that raising teenagers can be difficult. I'm glad you are taking your daughter for counseling as the cutting and behavior are a pretty clear sign that she has a lot going on in her head that she is unhappy about.
My husband nearly left because of the difficulties of raising my kids. He was feeling totally disrespected by my eldest and although I was too he does not have that unconditional love for my kids.
My eldest has gone into screaming rages when he doesn't get his way (for example, he wanted to borrow my car to visit his girlfriend at work during an ice storm where I wouldn't even go on the road) The things that come out of his mouth when he's like this are very hurtful and disturbing, but I stay calm and don't let him get me riled up (I'm not a yeller) and just keep repeating "you need to stop screaming if you want to talk to me"
My son has gotten into my face screaming and has physically pushed/hit me. Those are the times where my husband steps in and holds him back which makes him go off on my husband (who is totally non-violent) so it's a bad situation all around and my husband felt that he didn't want to live with this anymore and we could both see how it affected my 2 younger kids.
Here's what I did. I made a list of every instance I could think of over the years where he made a big (or small) impact on them. I talked to him about parenting and my perspective (being their birth mom) is that it's my responsibility to guide them into being healthy, happy, responsible adults and I always keep that in mind when they act up. I then told my husband the things he has done for my kids and the impact he has made and what it would do to them if he left (he really felt they would be better off) I talked about the hurtful things my son has said to both of us and that him not being his dad, etc. were said because my son was pulling the most hurtful things he could think of, not because he believes them...I was at the receiving end of 'you are a horrible mother and I hate you'. I told him parenting can feel very one-sided at times when you pour in your time, energy, money and get disrespect in return it's not always obvious that you are making any progress at all. Then I was at my son's band concert and a woman I did not know walked up and said 'are you his mom? I have to tell you, I drive your son's bus and I know how teenagers can be and you must be doing something right because he is the nicest, most respectful kid I have ever met' And I almost had to hold his arm and say 'you mean this one?' So the rewards may come at a much later time, and as adults we tend to avoid people that are 'takers' and that don't seem to appreciate us but when it's our own we need to go against that grain, which is natural for me but not for my husband as their step-dad.
He has always been good with my kids, taking them to the movies & never misses a game or concert, but he is now going a step further by just going to them and asking how their day was, having conversations about the latest with my eldest's car stereo & going out to look at what he installed & listening to the latest battle on WOW that my middle son had today or taking my daughter to batting cages before her next game. I'm sure he could think of many things he'd rather be doing but he's really stepped up to make sure they know he cares about them.
I can imagine how your husband is feeling. I hope when the dust settles a bit, you can have a good talk about what to do next. Make sure he knows he is valued, even if he can't see it right now. Kids will act up more when it comes to 'your car/your house/etc' because they are comfortable there. It's not an intentional attack on him. I hope you are able to get some family counseling as I don't think Plan B (living with their dad) is even an option from what you've said. You must be drained I'm sure (mom is the glue of the family in these cases, I know first-hand) but I would try to show a lot of empathy towards your husband and come up with solutions that will let him see there is a light at the end of the tunnel.
One thing I did say was that if my son could not control his temper, he would have to move out after high school. He has now arranged to live with his dad and go to college near him (which makes me nervous but I know it's the best considering our situation) He knows now that he is 18 I have no obligation to provide food/clothing/shelter if he doesn't live by our rules. We have had a very peaceful summer so far
My other suggestions were that my husband get more involved with the day to day parenting. Instead of mom taking care of everything, he will step in now and go with my daughter to buy her a new mitt, etc. so they see now that he's part of our family, not just mom's husband.
Anyway, hang in there....you are not alone....It will get better...don't give up hope.