As per my understanding I think:
Your third assertion is incorrect.
It's perfectly legal for two unequal objects to have the same hash code. It's used by HashMap as a "first pass filter" so that the map can quickly find possible entries with the specified key. The keys with the same hash code are then tested for equality with the specified key.
You wouldn't want a requirement that two unequal objects couldn't have the same hash code, as otherwise that would limit you to 232 possible objects. (It would also mean that different types couldn't even use an object's fields to generate hash codes, as other classes could generate the same hash.)