One 365 mailbox and several non 365 mailboxes

This looks like a mail flow loop rather than a connector validation issue. The error 554 5.4.14 Hop count exceeded typically means mail is bouncing back and forth between Exchange Online and your external IMAP mail server.

A few things to check:

• MX Record & Routing
  If your MX record points to Exchange Online, then O365 needs to know which users are hosted on IMAP vs. Exchange Online. For IMAP users, you’ll usually need a mail flow connector or smart host rule that routes their mail to your external server.
• Accepted Domain Setting
  Make sure your domain in Exchange Online is set as Internal Relay (not Authoritative). Authoritative tells O365 that all mailboxes are hosted there, which causes a loop when it can’t find the IMAP user.
• Connectors
  It sounds like you already created in/out connectors. Ensure the outbound connector from O365 → IMAP server is selected in your transport rules.
• Command Confusion
  You’re right: Start-Edge Synchronization applies only to on-premises Exchange servers. In Exchange Online, directory sync isn’t done this way. For your case, it’s purely a mail flow / accepted domain configuration issue, not an edge sync issue.

In short:

Set your domain in Exchange Online to Internal Relay.
Confirm your connector routes unknown recipients to your IMAP host.
Test delivery with Get-Message Trace in Exchange Online to see where it’s looping.

That usually resolves the loop in hybrid-style setups where some users are on O365 and others remain on IMAP.