The answer to question 2 is easy - yes you can use any Object you like. Maps that have String type keys are widely used because they are typical data structures for naming services. But in general, you can map any two types like Map<Car,Vendor>
or Map<Student,Course>
.
For the hashcode() method it's like answered before - whenever you overwrite equals(), then you have to overwrite hashcode() to obey the contract. On the other hand, if you're happy with the standard implementation of equals(), then you shouldn't touch hashcode() (because that could break the contract and result in identical hashcodes for unequal objects).
Practical sidenote: eclipse (and probably other IDEs as well can auto generate a pair of equals() and hashcode() implementation for your class, just based on the class members.
Edit
For you additional question: yes, exactly. Look at the source code for HashMap.get(Object key) calls key.hashcode to calculate the position (bin) in the internal hashtable and returns the value at that position (if there is one).
But be careful with 'handmade' hashcode/equals methods - if you use an object as a key, make sure, that the hashcode doesn't change afterwards, otherwise you won't find the mapped values anymore. In other words, the fields you use to calculate equals and hashcode should be final (or 'unchangeable' after creation of the object).
Assume, we have a contact with String name
and String phonenumber
and we use both fields to calculate equals() and hashcode(). Now we create "John Doe" with his mobile phone number and map him to his favorite Donut shop. hashcode() is used to calculate the index (bin) in the hash table and that's where the donut shop is stored.
Now we learn that he has a new phone number and we change the phone number field of the John Doe object. This results in a new hashcode. And this hashcode resolves to a new hash table index - which usually isn't the position where John Does Donut shop was stored.
The problem is clear: In this case we wanted to map "John Doe" to the Donut shop, and not "John Doe with a specific phone number". so we have to be careful with autogenerated equals/hashcode, because the might use unwanted fields and introduce trouble with HashMaps and HashSets.
Edit 2
If you add an object to a HashSet, the Object is the key for the internal hash table, the value is set but unused (just a static instance of Object). Here's the implementation from the openjdk 6 (b17):
// Dummy value to associate with an Object in the backing Map
private static final Object PRESENT = new Object();
private transient HashMap<E,Object> map;
public boolean add(E e) {
return map.put(e, PRESENT)==null;
}