How do I solve this "Cannot read property 'appendChild' of null" error?

The element hasn't been appended yet, therefore it is equal to null. The Id will never = 0. When you call getElementById(id), it is null since it is not a part of the dom yet unless your static id is already on the DOM. Do a call through the console to see what it returns.

Your condition id !== 0 will always be different that zero because you are assigning a string value. On pages where the element with id views_slideshow_controls_text_next_slideshow-block is not found, you will still try to append the img element, which causes the Cannot read property 'appendChild' of null error.

Instead of assigning a string value, you can assign the DOM element and verify if it exists within the page.

window.onload = function loadContIcons() {
    var elem = document.createElement("img");
    elem.src = "http://arno.agnian.com/sites/all/themes/agnian/images/up.png";
    elem.setAttribute("class", "up_icon");

    var container = document.getElementById("views_slideshow_controls_text_next_slideshow-block");
    if (container !== null) {
        container.appendChild(elem);
    } else console.log("aaaaa");

    var elem1 = document.createElement("img");
    elem1.src = "http://arno.agnian.com/sites/all/themes/agnian/images/down.png";
    elem1.setAttribute("class", "down_icon");

    container = document.getElementById("views_slideshow_controls_text_previous_slideshow-block");
    if (container !== null) {
        container.appendChild(elem1);
    } else console.log("aaaaa");
}

Just reorder or make sure, the (DOM or HTML) is loaded before the JavaScript.