How to set a specific starting volume for <audio> tag

Question 1

You're not getting the element, and the error shows up in the console

Cannot set property 'volume' of null

Move the script to right before the </body> tag, or do :

window.onload = function() {
    var backgroundAudio=document.getElementById("bgAudio");
    backgroundAudio.volume=0.5;
}

EDIT:

Nothing is more annoying than disabling right clicks, but anyway, here's what you have

;(function($){
mySong=document.getElementById("bgAudio");
mySong.volume=0.2;

setVolume = function(bgAudio,vol) {
    sounds[bgAudio].volume = 0.33;
}
})(jQuery);

now change it to

jQuery(function($) {
    $("#bgAudio").prop("volume", 0.2);

    window.setVolume = function(bgAudio, vol) {
        sounds[bgAudio].volume = vol;
    }
});

Question 2

Use JavaScript to set the volume, however you need to make sure to have the DOM ready to modify any objects. Perhaps use an ondocumentready event, or a Click event from a call-to-action.

First, get the video object you want to modify.

bgAudio = document.getElementById("bgAudio");

Then use the bgAudio object to set its properties, like the volume.

bgAudio.volume = 0.2;

Since you are working on a website, I would do something like this (assuming you can use jQuery):

$(document).ready(function() {
    // the site has now loaded, grab the video!
    bgAudio = document.getElementById("bgAudio");
    // now tweak the volume!
    bgAudio.volume = 0.2;
    // now, play it!
    bgAudio.play();
});

Check out this reference material on jquery ready, events and videos: