I have just one three year old, and I can echo that they can just be not very likeable and can get on your last nerve, baby or not. The not listening, the attitude, the drama, the "I don't like you" is just part of the three year old landscape. Add a new baby into the mix and it gets more intense.
I think it's fine that your daughter continues day care. If this is something that she is used to then I personally think it is good for her to continue her routine and her friends and her other stuff. She may feel jealous that baby gets to stay home, but I don't think her staying home with you would make the situation any better. She'd get less stimulation and less attention and you would be stretched even thinner.
I know it hurts to hear things like "I don't like you". In our experience (and this was my son saying that to my husband), the more you try to push yourself on them, the more they resist. What I try to say when my son says that to me is "It makes me sad to hear that, because I like you a lot and love you very much" and then just walk away and not force the issue. If daddy is available to put on pajamas, let him do it. If daddy is not available or able to do it, I say "I know you want daddy but he is busy (or whatever) now so I will help you". Kids change their "favorite" parent, and after a period of aversion to my husband they are now best buds, so it will pass.
I think even though you say she doesn't seem jealous, I am sure she is. It's a huge transition for her. You don't need to win her by doing happy fun outings. It would probably help for your husband to take over baby duty for a while and you spend a little time with your daughter. It can be something simple like reading her favorite book to her. Little every day things on a pretty consistent basis.
Their relationship will change over time too. Right now the baby is not a great playmate. Does not DO anything except lie there, poop, cry and suck all of mommy's time and attention. As baby grows more interactive, they will actually be able to play together.
Also, you really can't guarantee that sibs will be close. Even if you do everything right as a parent (if there is such a thing) they may be best friends, or they may not. They may be great friends as kids and then grow distant, or they may not get along all through childhood and grow super-close as adults. You don't have control over that part. You can demand that they treat each other with respect and consideration and empathy while they reside in your home, but the rest is up to them.
Good luck. I'm gonna be facing similar things in a few months.