Hello,

The problems are caused either by child theme customization or third parties plugins.

I have switched to the parent theme and disabled the third parties plugins: https://gyazo.com/a0e9aeab92d96fd20957ea43abef9212

The product categories menu appears: https://gyazo.com/796ad0a2e19ad6f455a59106e42dfef2

AJAX search works as it should: https://gyazo.com/fc274f01ceaaa71535a67f1796b1bdb6

The search result page shows the products in the proper way: https://gyazo.com/7e5dd07430a8624484bb6b47994ec2a5

Please switch to the parent theme, then deactivate the plugins in the same way as I did, then activate the plugins one by one checking the site to detect which one causes the problem.

If you have any questions please feel free to contact us.

Best Regards