@Gixertrixter I am facing the same problem. I followed all instructions from the manual and I got good feedback from other users, but still, I can only send emails, not able to receive any. My third party provider is telling me, that this has to be solved by Shopify, as they are not in charge anymore. I am lost right now. Can you please let me knwo how you solved it? Thanks a lot.
If that can be of help: if you're not able to receive emails it means the MX record of your domain DNS is not correct. That record tells what mail server handles emails for your domain, and it should point to whatever email service provider you're using.
Additionnal TXT records are recommended, especially SPF and DKIM which basically ensure only you is allowed to send emails from your domain email addresses.
Usually the DNS of your domain can be edited at your domain registrar website. If your domain registrar is Shopify (i.e. you acquired your domain through Shopify or you transferred it to Shopify), then you can edit your DNS on Shopify's DNS manager. If in the other hand your registrar is a third party (Namecheap, GoDaddy, ...), you'll need to edit your domain DNS there.
Please note that even if you point your domain to your Shopify website or "add/associate" your domain to your Shopify account, that doesn't mean your registrar is now Shopify. You'll still need to edit your DNS at your registrar website. This is a common source of confusion.
Finally, if you're not sure where your DNS is, you can check the NS record (or the SOA record, which is the Start of Authority) of your domain. There are many tools for that, for instance here. And you can find the registrar of a domain via whois. The NS record (and the SOA one) tells what DNS servers host your DNS records. And 99.99% of the time those DNS servers are provided by the domain registrar from which you acquired your domain.
Hope this is somewhat helpful!