How to write asynchronous functions for Node.js

You should watch this: Node Tuts episode 19 - Asynchronous Iteration Patterns

It should answers your questions.

Just passing by callbacks is not enough. You have to use settimer for example, to make function async.

Examples: Not async functions:

function a() {
  var a = 0;    
  for(i=0; i<10000000; i++) {
    a++;
  };
  b();
};

function b() {
  var a = 0;    
  for(i=0; i<10000000; i++) {
    a++;
  };    
  c();
};

function c() {
  for(i=0; i<10000000; i++) {
  };
  console.log("async finished!");
};

a();
console.log("This should be good");

If you will run above example, This should be good, will have to wait untill those functions will finish to work.

Pseudo multithread (async) functions:

function a() {
  setTimeout ( function() {
    var a = 0;  
    for(i=0; i<10000000; i++) {
      a++;
    };
    b();
  }, 0);
};

function b() {
  setTimeout ( function() {
    var a = 0;  
    for(i=0; i<10000000; i++) {
      a++;
    };  
    c();
  }, 0);
};

function c() {
  setTimeout ( function() {
    for(i=0; i<10000000; i++) {
    };
    console.log("async finished!");
  }, 0);
};

a();
console.log("This should be good");

This one will be trully async. This should be good will be writen before async finished.

You seem to be confusing asynchronous IO with asynchronous functions. node.js uses asynchronous non-blocking IO because non blocking IO is better. The best way to understand it is to go watch some videos by ryan dahl.

How do I write asynchronous functions for Node?

Just write normal functions, the only difference is that they are not executed immediately but passed around as callbacks.

How should I implement error event handling correctly

Generally API's give you a callback with an err as the first argument. For example

database.query('something', function(err, result) {
  if (err) handle(err);
  doSomething(result);
});

Is a common pattern.

Another common pattern is on('error'). For example

process.on('uncaughtException', function (err) {
  console.log('Caught exception: ' + err);
});

Edit:

var async_function = function(val, callback){
    process.nextTick(function(){
        callback(val);
    });
};

The above function when called as

async_function(42, function(val) {
  console.log(val)
});
console.log(43);

Will print 42 to the console asynchronously. In particular process.nextTick fires after the current eventloop callstack is empty. That call stack is empty after async_function and console.log(43) have run. So we print 43 followed by 42.

You should probably do some reading on the event loop.