Your daughter is expressing her independence with you. This is a normal phase. She doesn't have to do that with her father because he's not around very often. The fact that she seems to be overdoing this independent phase is probably related to how much she actually does love you. If she didn't love you she wouldn't have to tell you that she doesn't love you.
Of course it hurts! It most likely hurts you more than you'd expect because you're dealing with some very heavy relationship issues. You are facing the possible loss of three other extremely important people in your life. Even tho they are alive they are not the same healthy people you've grown up with. When your daughter pushes you away that's a fourth loss.
Also remember that your daughter is facing loss too. Even if you haven't told her about the illnesses she knows something is different. And having her dad gone most of the time is also a loss. Her bringing things for him could be indicative of her effort to hold onto him. And her not bringing things for you could be indicative that she's letting go of you because you might get sick and go away too. She needs lots of reassurance of your love and knowledge that you will always take care of her.
I adopted my daughter as a special needs child and know from my experience that I felt more hurt from her saying, "I hate you" than I do now when my grandchildren say that. Intellectually I know that I'm just as much her mother as if I'd given birth but there is a part of me buried inside that feels insecure. I also know that there is a large part of my daughter that feels insecure. It's like I chose her. I can unchoose her. I don't know if your daughter knows she's adopted or if a preschooler would be aware enough to even feel that unconsciously. My school age daughter was not consciously aware but our therapists said she had the sense buried deep inside that I wasn't her "real" mother. (I do hate the term "real" mother.) I was aware of feeling at times, especially when she was saying, "I hate YOU, over and over, that my own sense of rejection was over the top. She brought things home from school for her birth mother. That hurt. I don't know if that applies to you at all.
My grandchildren, who are now 6 and 9, still say that they hate me every once in awhile. In counseling I learned that what is most often happening, at that age anyway, (that is the same age that my daughter came to live with me) is that they're unhappy about something and it just feels right to them to project that unhappiness on me. They don't actually mean that they hate me. They mean that they're having a "bad day" and don't want to talk about it.
I learned with my daughter and am now doing the same with my grandchildren to say, "oh, that's too bad. I still love you." Or I act as if I didn't hear what they said. Or I say, "oh, that's too bad." and nothing more. I never make their words into a big deal. With my daughter I had to learn how to stop showing her my pain most of the time. I did this by giving myself pep talks at times when I was away from her as well as the times she said the words. Remind yourself that this is a phase, that you know she loves you even tho it doesn't feel like she loves you.
The next thing I did was focus more energy on myself and doing things that helped me feel better. I was in counseling which helped tremendously. I recommend that for you. You have really unhappy things happening in your life that are bringing you down. And you don't have your husband's consistent presence to help deal with any of it. I also began taking an anti-depressant which I continue to take off and on. I was raised by a chronically depressed mother which makes me prone to depression.
Depression makes it very difficult to deal with your daughter. You need someone telling you that they love you and giving you hugs.
You could try talking with your daughter, when you are both calm and getting along, telling her that you're feeling very sad and need hugs. Most of all tell her that you love her even when she says she doesn't love you. Ask for her help in making home happier.
Then work very hard at not responding to her when she says hateful things or brings home something for her father. Be very matter of fact. If you can smile and respond with "I love you anyway. You can never make me not love you." Exclaim over that paper for her dad. Tell her he'll love it. That sort of thing. Put it some place special for dad. By doing that you're making yourself a part of the "gift.:
Play the part of the unflappable adult until you are the unflappable adult. She needs to know that she can't hurt you with words that she says every day. That gives her too much power. Being able to control your emotions is frightening to her and probably causes her to be even less loving.
She already knows by now that her words and actions hurt you so you can stop telling her. Find a way to stop most of the hurt. When you're not feeling hurt tell her that her words do hurt. Provide a consequence when she's disrespectful. But remember expressing a feeling with words is not disrespect even when it hurts.
It'll be difficult but when she gets angry when you try to tell her something or to help her back off. Say something to the effect, "I think you're angry. I can't help you when you're angry." If you're trying to help her do something that she needs to do, such as brushing her teeth, eating breakfast, going to bed, repeat that you still expect her to finish and then leave the room. If she's getting ready for bed, leave the room after telling her to call for you when she wants some help or when she's ready for a story or whatever the next step is. If she doesn't get ready for bed give her a consequence.
If you haven't already, tell her what the consequences are for misbehavior. Again it's a consequence for not doing what you've asked her to do. It's not a consequence for expressing her feelings in words. If the words include swearing and you don't allow swearing in your house (which means you also don't swear) then provide a consequence for that. Be as specific as you can. You'll get better at it.
Have someone with whom you can express your feelings. At this point the goal with your daughter is to help her be able to be independent from you without having to say mean things. The goal for you is to learn how to feel good about yourself even when she does say mean things. You take back your power. It sounds like she currently holds the power in these situations.
I do know how painful hearing these words are. Be gentle with yourself. You can't just turn off your hurt. You can redirect your focus. For me the best way to do that was to have friends that I could call, later, who would tell me I was doing a good job and who would then help me laugh.