Updates now causing 404 and ajax errors

I recently carried out a little bit of work on a friends website adding in some text and images and editing button labels etc. All of that has gone well. Unfortunately, I suggested that we update the WP core and Plugins as they were out-of-date and everyone recommends keeping them up to date. I took a backup and then updated WP from 3.3.2 to 4.3, WP Ecommerce from 3.8.8.3 to 3.9.5 and the Gold Cart plugin is running at 2.9.7.

This has resulted in breaking the front of the site although the back end appears to be working OK. When you try and add an item to the cart, the loading icon just hangs forever and I can see I now have 404 messages generated within Firebug such as;

http://thebigheadboard.co.uk.gridhosted.co.uk/products/headboards/giraffe-cream/index.php?ajax=true

even though you can actually browse to the file. I’ve also spotted a couple of instances of CORS blocking with the admin-ajax.php file because it’s referencing it over https when the site is only hosted over http. From what I can gather, this may be a problem with the variable ajaxurl, but I can’t find where this is set at all.

I’ve literally spent hours trying to find out what the cause of the problem(s) is/are and how to fix it/them but to no avail. Can anybody provide some pointers please? I’ve tried the usual tests to rule out issues with the plugins and I’ve tried saving the Permalinks settings several times in case these somehow got screwed up. No luck so far.

Thanks in anticipation.