I added a small snippet of custom CSS to your product template as a workaround to stop the sidebar from displaying beneath the content (except on smaller screens).
It could be improved upon, but should help in the meantime until the escalated issue is resolved.
You'll need to try a similar CSS workaround as on the single product pages. I haven't checked if the markup issue is the same on the archives as it is on the single product pages, you may just be able to copy across the custom CSS to your archive template, or it may need tweaking when you do.
Actually, checking, you are not using a custom archive, the standard archives are being displayed.
So, I changed the setting at Toolset > WooCommerce Views to say use the standard WC product archive template instead of the Toolset one (which is not needed if you are not customising the archive) and that has fixed the sidebar issue on the archive pages.
Is it only happening on the production site but is working okay on the staging site?
I visited the production site again and can see the problem. We could try de-activating and re-activating plugins as I initially described, but it would be better to get to the bottom of the issue if it has re-surfaced.
So I was going to take a fresh copy of the site which I'd be able to pass to my colleagues for some more testing—but, having installed the All in One WP Migration plugin on your site, it won't activate.
It seems like you maybe have something else going on on your site.
Thanks for the error logs, I've tidied it up somewhat to isolate the problems so that you can see them:
[Thu Jun 21 07:11:14.139637 2018] [:error] [pid 24650] [client 5.148.67.47:63111] PHP Fatal error: Call to a member function is_type() on boolean in /nas/content/live/flyingtiger/wp-content/plugins/woocommerce-pre-orders/includes/class-wc-pre-orders-product.php on line 257, referer: <em><u>hidden link</u></em>
[Thu Jun 21 08:21:02.128478 2018] [:error] [pid 21846] [client 92.40.248.162:32827] PHP Fatal error: Call to undefined function is_product_category() in /nas/content/live/flyingtiger/wp-content/uploads/dynamik-gen/theme/custom-hook-boxes.php on line 91, referer: <em><u>hidden link</u></em>
[Thu Jun 21 08:21:08.474096 2018] [:error] [pid 22129] [client 54.197.115.34:32945] PHP Fatal error: Call to undefined function is_product_category() in /nas/content/live/flyingtiger/wp-content/uploads/dynamik-gen/theme/custom-hook-boxes.php on line 91
[Thu Jun 21 08:21:09.134661 2018] [:error] [pid 22451] [client 54.197.115.34:32969] PHP Fatal error: Call to undefined function is_product_category() in /nas/content/live/flyingtiger/wp-content/uploads/dynamik-gen/theme/custom-hook-boxes.php on line 91
[Thu Jun 21 08:21:21.573768 2018] [:error] [pid 22760] [client 92.40.248.162:33127] PHP Fatal error: Call to undefined function is_product_category() in /nas/content/live/flyingtiger/wp-content/uploads/dynamik-gen/theme/custom-hook-boxes.php on line 91, referer: <em><u>hidden link</u></em>
[Thu Jun 21 08:21:23.019329 2018] [:error] [pid 22786] [client 92.40.248.162:33156] PHP Fatal error: Call to undefined function is_product_category() in /nas/content/live/flyingtiger/wp-content/uploads/dynamik-gen/theme/custom-hook-boxes.php on line 91, referer: <em><u>hidden link</u></em>
[Wed Jun 20 15:51:36.927944 2018] [:error] [pid 17506] [client 81.158.223.32:38769] PHP Fatal error: Call to a member function is_type() on boolean in /nas/content/live/flyingtiger/wp-content/plugins/woocommerce-pre-orders/includes/class-wc-pre-orders-product.php on line 257, referer: <em><u>hidden link</u></em>
You can see that your theme is creating fatal PHP errors (related to custom-hook-boxes.php) and also the plugin woocommerce-pre-orders is likewise creating fatal errors.
Errors from one plugin can have unexpected knock-on effects on code elsewhere, so while you have these errors on your site it is not possible to say why the Toolset-related problems occur. In particular, the non-appearance of the custom fields on the product edit pages may be linked to either one of these other plugin errors.
I'm going to try and test the WooCommerce pre-orders plugin to see if I can reproduce this error on a local test site, but in the meantime I suggest you test your site with that plugin disabled and using the parent Genesis theme rather than the dynamik theme (which you can do on your staging site).
If you still can't see the product custom fields then the cause must lie elsewhere, but if they work okay then the errors reported for the theme or that plugin will be responsible.
The product category archives are not being displayed by Toolset.
In Toolset > WooCommerce Views you'll see that the template used for product archives is the standard WooCommerce template and not the Toolset WooCommerce Views template, so what is output on the product category archive pages is determined by your theme and WooCommerce.
To double-check I temporarily disabled the Toolset WooCommerce Views plugin and the Views plugin so that Toolset can't be having any influence on how the product archives are rendered and it didn't make any difference.
