Greetings ptkmac,
It sounds as though you’re having a couple of issues with Messages on your Mac. To confirm, you’re having issues with notifications for incoming Messages not showing on your Mac, and your contact names aren’t showing in Messages. Apple Support Communities is a great place for help!
With Messages on your Mac, have you verified the notification settings as outlined in the following article? If not, verify the notification settings are correct.
Use notifications on your Mac
If the notification settings are correct, do you notice if notifications come to your Mac first after you’ve opened and started using Messages on Mac versus on your iPhone? For example, if you send or receive a message on your Mac, do the following incoming messages appear on the Mac first before showing on your iPhone?
With the contacts issue, are your contacts synced to iCloud or an email account? If so, have you verified that contacts are selected to sync in System Preferences > Internet Accounts (for your emails), or System Preferences > Apple ID for iCloud? If not, make sure your contacts are syncing, and appear in the Contacts app on your Mac. For steps on how to set up iCloud Contacts and troubleshoot syncing issues with iCloud Contacts, take a look at the following articles.
Set up and use iCloud Contacts
If your iCloud Contacts, Calendars, or Reminders won’t sync
Let us know how that goes!