C.,
9 mo is a VERY common age for kids to start separation anxiety. A crawling baby has come to discover that she is a separate person from you, and although she's happy to have her increased freedom, she's also a little nervous about it. She will grow out of it with your support. When you leave her at care, give her a reassuring hug and kiss, be brief, tell her you'll be back, and go. The more you hesitate, the more she'll wonder if there's something to worry about. If you are confident, she can be, too! If she melts when you get there, hug her and tell her how much you missed her, too. Now you're giving her words for what she's feeling. At home, try wearing her in a sling or carrier so she can have lots of "face time" (wear her facing you) while you get things done. This, too, shall pass.
As for the biting- she's not really old enough to understand that it's hurting someone else when she does it. It's possible that she's teething and it feels good. If she's an "oral" baby (loves to chew, bite, and mouth everything!), she may bite when she's frustrated to relieve stress. Give her an alternative- a chewy toy or a washcloth. Be sure her care provider understands that biting in little ones this young is NEVER a mean act. It can be frustrating, yes. And mortifying for the mom of the biter. But these things happen as children explore personal boundaries and learn what's safe and what's not. My own dd has bitten me out of frustration, and I make a very sad face, tell her that it hurt me, we bite food and chewies, not people. She will learn.
And as for the tantrums- they all have 'em! It's her way of expressing that she's upset is all. Remember that your little one has no other way of telling you how she's feeling- no words for mad, hurt, upset, hungry, etc. So until she learns them, she will use body language to tell you. As she's flailing, tell her what you see. "I know you're angry, and it's hard to wait your turn." etc. It may feel silly at first, but you're teaching her how she's feeling so she can tell you later. You could also start teaching her sign language at this age. Babies can sign almost a full year before they can speak, so it will give her a way to tell you all sorts of things (my dd has a vocab of 15 signs at 1yr old!).
You're doing a great job as a mom. These challenges will come and go, and just when you're done with one stage, the next begins! Remember, you are the perfect mother for your child, and you will grow and learn about eachother together!
If you'd like more info on baby sign language, feel free to send me an email or message and I'll let you know what I used.