I don't think mutability is a concern if your goal is to get a bidi map between strings and (mutable) _instances_; i.e. if you just want to use the reference to the dict itself as the hash rather than the contents of the dict itself.
It’s been a while, but I’m pretty sure the reason why a dictionary can’t have a dictionary as it’s key is because the dictionary class doesn’t implement the __hash__ method that would compute the hashcode.
Sure, but as others have mentioned, you can make your own choice by subclassing dict and defining your own hash function, as long as that function is defined in a way that follows the appropriate rules.