Hikashop's notification emails do not arrive in the customer's gmail account

Hi,

Do the emails arrive for users not using gmail ?

First, you can check that the emails are sent by HikaShop by looking in the Customers>Emails history menu.
I suppose you'll see them there, which means that HikaShop sent the emails to the email server of your hosting to be sent to the receiver. If the sending of the email fails, the hosting server won't notify HikaShop back, however, it will log the issue in the log file of the email server. So, the hosting should be able to tell you what is the problem.
Also, if it's only for some emails, then it's likely a spam issue. In that case, I would recommend using www.mail-tester.com/ to check the state of your email setup. That will tell you if some things are not configured properly for the emails.