"Is a bean allowed and equals and hashCode method?"
Yes, a bean is just a class, just like any other class, so it can have an equals and hashCode method - in fact it might even be a good idea to provide these methods.
"Do all fields in a bean have to have both getters and setters?"
No, only the fields that you want to expose to the world outside the bean. The bean might have fields that are for internal use only. And indeed you can have read-only properties for which you only provide a getter and no setter.