Tell me about your baby's peanut butter reaction

I gave my 6 month old baby peanut butter on Sunday (thinned out with water on the end of a spoon). She pretty immediately developed red splotches around her mouth. I took her to the hospital and got a coffee and watched her, but they faded and she developed no other symptoms, so we skipped the ER and just went home. I'll be talking to her pediatrician tomorrow, but if this happened to you, please let me know what happened next!!!! Some things I'm wondering include:

• Can I have her try other top allergens (egg, dairy) or should I wait?
• How long is too long to wait before seeing an allergist? Is it better to get desensitization going ASAP?
• Do we need to get rid of peanut product in our house?
• It was so mild overall - is she definitely allergic?

I'm really stressed and disappointed by this development and craving more information and experiences!!!!