If by nature she is sensitive you cannot change that without asking her to stop showing her feelings and eventually ending up with further problems, I don't know what the school hope to achieve by her seeing a psychologist but it won't make her less sensitive. She is too young to reason things out and understand that what she feels may be OTT. What can be done is finding her a way to express herself better. To acheive that you and her teachers have to accept her feelings as they are now, they need acknowlegdement and reassurance, once she gets that all round she'll feel secure about her feelings and the need to express herself so dramatically will reduce because she's being listened to. ATM it sounds like school are pressuring her into changing the way she feels and that will only cause more frustration to her. That doesn't mean when she screeches I hate you at you that you just accept it, but work towards changing the way she expresses it rather than the feeling itself. At 4 hating someone means at that moment there cross with you for something you've done (most commonly saying no to them!) so don't take it literally...and they ALL do it!

Has she always been like this or is it a sudden change in behaviour? Could there be an external factor, maybe someone picking on her in school or an an adult thats being mean to her behind your back?

If by spoiling a child you mean with toys and always saying yes to them thats easily done, but there is no way you can spoil a child with cuddles, time and respect for there feelings. Give her as much as you are able to and enjoy it!