Should I be able to see future messages from 'archived' dogs/families in my regular inbox?

Yes, if I recall correctly, there’s a small circle above inbox tab notifying of message. Sometimes, these messages might not be related to another booking but rather something else. If it’s related to another booking, then yes move it into your inbox.You can move it back-and-forth from Archive to Inbox and vice versa as much as needed.

Addition: It might be a good idea to go to your Settings, from there Notifications, and check which are green/turned on versus gray/turned off. You can choose to be alerted by text by email to all new messages or only specific inquiries, bookings, etc.

I have all my requests come through as text messages also. I have booked previously archived requests this way in the past and another one just this week.