Equality and Coercion could be harmful

The goal of this page is to gather all problems due to equality accepting coercion in borderline cases:

sage: bool(pi == 3.14159265358979323)
True
sage: hash(pi)
2943215493
sage: hash(3.14159265358979323)
1826823505

But gathered from Python 2.7 documentation:

object.__hash__(self)
Called by built-in function hash() and for operations on members of hashed collections including set,
frozenset, and dict. __hash__() should return an integer. The only required property is that objects
which compare equal have the same hash value;

As a first consequence of the above behavior:

sage: {3.1415926535897932: 'approx', pi: 'exact'}
{3.1415926535897932: 'approx', pi: 'exact'}
sage: {0:"exact", 0.0000000000000000000:"approx"}
{0: 'approx'}

And also:

sage: pii = 3.14159265358979323
sage: bool(pii == pi)
True
sage: dd = {pi: "exact"}
sage: pi in dd
True
sage: pii in dd
False
sage: pii in dd.keys()
True

sage: F1 = FiniteEnumeratedSet([0.000000])
sage: F2 = FiniteEnumeratedSet([0])
sage: F1 is F2
True
sage: F2.list()
[0.000000000000000]
sage:

sage: S = SymmetricFunctions(QQ)
sage: x = S.s()[5]
sage: y = S.p()(x)
sage: x == y
True
sage: hash(x), hash(y)
(-1840429907820881728, 5178019317311573726)