Hey all, I got this worked out — well, rather someone who understands computers far better than I got this worked out. As best as I can understand, this was the problem:
wc-ajax (WooCommerce ajax) was returning a 404 error for the following WooCommerce calls:
wc-ajax=get_refreshed_fragments
wc-ajax=update_order_review
Also there were some odd URL Rewriting rules in htaccess.
Sorry I can’t tell you more about exactly how to fix this problem. But this was the basics of it, I am told.