Hi Marc!
1. Well, it is very hard to tell, but based on these two code sections, I would say it is probably the first one. I would try changing this line:
<div class='search-wrapper'><?php upstore_get_search_form_by_category(); ?></div>
..to this:
<div class='search-wrapper'><?php echo do_shortcode('[wpdreams_ajaxsearchpro id=1]'); ?></div>
If that is the correct file, then this should do the trick.
2. The SKU should be stored within the _sku custom field, make sure that is selected for searching/indexing within the plugin options. The GTIN must have a similar custom field, so make sure to search for that one as well. That should resolve that problem.