According to MSDN, a hash function must have the following properties:
If two objects compare as equal, the GetHashCode method for each object m
A) You must override both Equals and GetHashCode if you want to employ value equality instead of the default reference equality. With the later, two object references compare as equal if they both refer to the same object instance. With the former they compare as equal if their value is the same even if they refer to different objects. For example, you probably want to employ value equality for Date, Money, and Point objects.
B) In order to implement value equality you must override Equals and GetHashCode. Both should depend on the fields of the object that encapsulate the value. For example, Date.Year, Date.Month and Date.Day; or Money.Currency and Money.Amount; or Point.X, Point.Y and Point.Z. You should also consider overriding operator ==, operator !=, operator <, and operator >.
C) The hashcode doesn't have to stay constant all through the object lifetime. However it must remain immutable while it participates as the key in a hash. From MSDN doco for Dictionary: "As long as an object is used as a key in the Dictionary<(Of <(TKey, TValue>)>), it must not change in any way that affects its hash value." If you must change the value of a key remove the entry from the dictionary, change the key value, and replace the entry.
D) IMO, you will simplify your life if your value objects are themselves immutable.