Hi,
I've tested on my end and with Joomla 3.8.7 and HikaShop .3.4.0, the terms and conditions checkbox works fine.
I've tested it on verzevoul's website and I could reproduce the issue. So it must be something linked to his setup. We'll, need a FTP access (or cpanel) to be able to debug the issue as we don't have it on our end.
Regarding your case RosenMcStern, if you have the same issue with the checkbox of the terms and conditions, then no need to open a new thread for that. However, regarding the add to cart issue, I doubt it has any link to the terms & conditions issue. So yes, it would be great to open a new thread, with a link to reproduce the issue and instructions if necessary.