Non-existing XML node breaking my script

You should separate your concerns better. Resist the temptation to write a function that does everything.

First concern: Request proxying

Let's abstract the plumbing for a proxied request into a function, complete with URL parameters that we can pass as an object.

function proxy(url, params) {
    if (!params || typeof params !== "object") params = {};
    return $.ajax({
        type: "post",
        url: "proxy.php",
        data: {address: url + "?" + $.param(params)}
    });
}

Remember to set the Content-Type header in proxy.php equal to the content type of the target page.

Second concern: Getting data from the remote API

We want an array of flights, where each flight has the properties schedule_time, flight_id, airport, airline, gate.

On top of that we want to pass in the query parameters (TimeFrom, TimeTo, airport, direction, lastUpdate) in a structured way.

function getFlights(params) {
    return proxy("http://flydata.avinor.no/XmlFeed.asp", params).then(function (xml) {
        return $(xml).find('flight').map(function () {
            var $flight = $(this);
            return {
                schedule_time: $flight.find("schedule_time").text(),
                flight_id: $flight.find("flight_id").text(),
                airport: $flight.find("airport").text(),
                airline: $flight.find("airline").text(),
                gate: $flight.find("gate").text()
            };
        }).toArray();
    });
}

Using .then() allows us to determine the overall result of the Ajax request, in our case it will be no longer XML but an array of uniform objects (no matter whether certain elements were missing in the XML).

Third concern: rendering

We want to turn an array of flights into a table. I would recommend handlebars.js for HTML generation, but we can do it manually with jQuery as well:

function renderFlights(flights) {
    var $table = $("<table>");
    $table.append("<tr><th>Departue</th><th>Flight</th><th>To</th><th>Airline</th><th>Gate</th></tr>");
    $.each(flights, function (i, flight) {
        var $tr = $("<tr>").appendTo($table);
        $("<td>", {text: flight.schedule_time }).appendTo($tr);
        $("<td>", {text: flight.flight_id }).appendTo($tr);
        $("<td>", {text: flight.airport }).appendTo($tr);
        $("<td>", {text: flight.airline }).appendTo($tr);
        $("<td>", {text: flight.gate }).appendTo($tr);
    });
    return $table;
}

Building HTML this way is safer than concatenating strings. It protects you against cross-site scripting attacks and random breakage due to unexpected special characters in the text.

Fourth concern: User interface wire-up

Now we can bind a button click very easily like this:

$(function() {
    $('#load_btn').click(function () {
        getFlights({
            TimeFrom: 1,
            TimeTo: 7,
            airport: "OSL",
            direction: "D",
            lastUpdate: "2016-04-12T15:03:00Z"
        }).done(function (flights) {
            var $table = renderFlights(flights);
            $("#result").empty().append($table);
        });
    });
});