I'll give you my thoughts on each area separately.
The hamster-- she is just too young to have intentionally killed this hamster. She hit him out of frustration-- sounds like she was being truthful when she said he would not listen. But don't think that this is because she has been hit herself or observed violence or anything, Frankly, a 3 year old can hit when frustrated. If she is not hitting other children in day care, then sounds to me like she is doing fine. On a related note, 3 is too young to be left alone with animals -- especially small ones like hamsters and mice-- for this very reason.
Terrible immune system-- I'm just not sure what you mean by this. If you mean she gets sick all the time, a lot of children do that-- so maybe you mean she has a specific immune condition? Just not sure about that.
Tantrums-- There is a book - How to Behave So Your Toddler Will Too, and How to Behave so Your Children Will Too, that offers good advice about developing a systematic behavior plan that will help you deal with the tantrums and other behavior issues. Another good book is SOS Help for Parents. If you employ a consistent behavior management strategy, with rewards and consequences for behavior, and follow it consistently, and have the day care follow it too, you will probably notice an improvement within a month.
Rocking-- many children rock as a comfort mechanism. You mention you had post partum depression. I had depression too, and grew up with a depressed mother. Children will develop self soothing mechanisms , some of them like rocking or maybe pulling hair ( which is what my son did)-- when they have to. When you're a depressed Mom, the children can tell -- and they keep trying, bless their hearts, to do the right thing, I think-- and will pick these things up as needed. If she displays no other social issues or developmental issues, then this type of behavior will gradually wane as she gets older I believe. RIght now, shes at a frustrating age, so she may do this as habit. If you have serious concerns about it, I would take her to a developmental pediatrician, just to get a second opinion.
How much time do you get to spend together one-on-one? Is this play-time, or daily grind stuff like housework, meals and bathtime?
How do you handle discipline at home and in public? What triggers the tantrums?