Solidity – Handling BigNumber Returned Values

Because your greet() function is marked constant, web3 by default simulates the call and returns the value, but no transaction is sent, and thus no state changes are retained. This behavior can be forced in non-constant methods using contract.method.call().

To call the function in a way that does send a transaction and update the chain, use greeter.greet.sendTransaction({from:eth.coinbase, gas:100000}).

Unfortunately, while other contracts will recieve the proper return value, web3 will return a transaction hash, as opposed to a return value. In this case, it makes sense to use Events.

For example, the solidity would look something like:

contract greeter { 
    string greeting; 
    uint calls; 
    event Greet (string greeting);
    function greeter ( string _greeting ) public { 
        greeting = _greeting; 
        calls = 0; 
     }
     function greet ( ) public returns ( string ) { 
         calls = calls + 1; 
         Greet(greeting);
          return greeting; 
     } 
 }

And to get a return value:

 greeter.greet.sendTransaction(
    {from:eth.coinbase, gas:100000},

    function (error, result){ 
    var event = greeter.Greet()
    event.watch(
        function(error, result){ 
             if (!error) console.log(result.args.greeting); 
         }
    );
)

Question : If I deploy my smart contract to the real Ethereum blockchain, then is there any user interface with which I could access the contract functions in the same way as I can do in MyEtherWalet?

You may use your node's console interface, like this:

<path-to-geth>/geth attach <path-to-datadir>/geth.ipc console

Console provides you with javascript interface, where you can manipulate with your node's state, call api methods, lookup blocks, account etc.

Question 2: What is the role of Web3 here?

Web3 is a ethereum client library. It can be written in different languages. Most popular (I think) is javascript version of Web3. Actually, using a geth's console you are using all the functionality of web3.