How to write asynchronous functions for Node.js

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.

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.