Hi,
Your link doesn't work:
monosnap.com/file/Qf63lDGEFg3yMzt4U9yaDfRIIwtMlT
First, switch to the default template of Joomla and see if you get an error message (or if it works). The contact form of HikaShop not working could come from a javascript error on the page (often coming from a conflict between javascript libraries, like the one in the template). Also, some template can not display error messages which could explain why you wouldn't get any message when submitting the contact form.
Also, go in the menu Customers>Emails history and check whether the email is logged there. If yes, then the email was sent by HikaShop. It could rejected by your email server for several reasons, the most likely being flagged as spam.
You can check the spammability of your emails with
www.mail-tester.com/