How can I Shorten and Simplify Ajax callback processes?
Here is a jsfiddle where I cleaned it up for you

I did not actually test it so there might be a syntax error here and there but its how you would do it overall.
It might not be 100% but it should get you started as far as the techniques you should look into such as:


Module pattern to create reusable code
Callback or options to support the ability to pass in a simple callback or a more complex object
Defaults with $.extend 
Functions returning callbacks


That's a lot of stuff and a bit overkill on this example but I wanted to showcase the techniques for you. Feel free to ask any questions.

A couple additional notes


You generally want to use the expression context parameter to limit your selectors
You probably want to avoid using ids and use classes instead since things get wierd if you accidentally end up with 2 of the same id on the page. This is something that happens more often as you start to make your views more composable. This is something that using the expression context can help with.
submit is only available on forms anyways, you could denote that explicitly $('form#add_question_member')
Do you really want to be returning false?
Consider using the form's action and method properties instead of hardcoding them - $(this).prop('action'), $(this).prop('method') at the same level as you serialize the form

//****This part can be in a seperate script file **********//
(function() { //use an immediately executing function to limit scope
    var submitFormDefaults = { //set up all the defaults to be used the function
        type: 'post',
        url: '/posts_in.php',
        dataType: "json",
        myAppUnWrapData: true //a custom option to unrwap your returned data by default
    };

var standardSubmitForm = function(successFnOrOptions) {
    // This allows your input to be just the success function or a richer object
    var options = $.isFunction(successFnOrOptions) ? {
        success: successFnOrOptions
    } : successFnOrOptions;

    // return the actual submit handler
    return function() {
        var op = $.extend(true, //deep clone
          submitFormDefaults, //use defaults unless overridden
          { data: $(this).serialize() }, //not in the defaults cause it needs access to 'this'
          options //any overrides specified
        );
        if (op.myAppUnWrapData &amp;&amp; op.success) { //if unwrap data custom option is truthy and there is a success callback
            var _oldSuccess = op.success;
            op.success = function(data) {
                _oldSuccess(data[0]);
            }
        }
        $.ajax(op);
        return false;
    }
}

//Export the local function to the gloabl namespace using Module pattern
window.MyApp = window.MyApp || {}; //create a global MyApp namespace
$.extend(window.MyApp, { //add some global functions. I prefer to use $.extend rather than directly assigning it - I have no good reason for this
    submitForm: standardSubmitForm //export our function to the global namespace           
});

window.MyApp.submitForm.defaults = submitFormDefaults //export the defaults as a global property on this function



})(); //Invoke the above immediately invoking function

//******************** Elsewhere ********************************//
$("#add_question_member").submit(window.MyApp.submitForm(function(response) { //this will return a function handler
    if (response !== 1) {
        $("#msg0").stop().fadeIn("slow").delay(8000).fadeOut("slow", function() {
            $("#entry_area").stop().fadeIn("slow");
        });
    } else {
        $("#msg1").stop().fadeIn("slow").delay(8000).fadeOut("slow", function() {
            $("#entry_area").stop().fadeIn("slow");
        });
    }
}));

// or...
$('form.blah').submit(window.MyApp.submitForm({
    url: '/somewhere/else',
    success: function() {
        console.log('ok', this, arguments);
    }
});

// or...
window.MyApp.submitForm.defaults.action = 'GET'; //changes it globally
$('form.blah').submit(window.MyApp.submitForm({
    url: '/somewhere/else',
    myAppUnWrapData: false,   //keep it from doing the data[0] thing
    success: function(data) {
        console.log('ok', this, arguments);
    }


});