It does. 'The desktop app is a native client for Mac and Windows with the full functionality of Yammer. Along with streamlined log in and SSO support, the app integrates with native operating system capabilities such as notifications, shortcuts, and launch on startup'
I couldn't say. I would potentially look to reinstall perhaps and see if it corrects. I would also check your yammer settings to ensure it is using Office 365 sign in credentials (this is through the Yammer Admin Page) Other than that, I would say it may be a ticket for support. I would imagine the logs are accessible either through explorer or via right clicking on the systray app (typically shift + right click). Unfortunately their doesn't seem to be a great deal of info out there on this
An alternative may be to wait for a bit - Yammer is getting much tighter intergration to Teams so you could use Teams and pinning Yammer in the app bar (left rail).