Facebook Messenger questions

First: I understand that Facebook is not an Apple product, even though I'm using it on my Mac. However, maybe someone can offer some advice since it's a widely used app. I'm having problems with one particular contact, so maybe it's not an FB problem, but it's an odd situation. An acquaintance who had an appointment with me didn't show up despite confirmation, and she didn't follow-up by text or phone as she usually would have. The phone number on her confirmation text wouldn't work. A week later I found a couple of messages from her via Facebook Messenger (sent the day she missed the appointment), including the notification that she had lost her phone and didn't remember any of my contact info. I've replied by Messenger several times, and the messages have been "sent," but she hasn't gotten back to me (since I don't know her new phone number, I can't text or call). Apparently there's a distinction between "sent" and "delivered," but I don't know what the status is in this case. Also, I occasionally see a message that my browser (Safari) isn't supported, and that I need to install a Messenger for Mac app. But just today I exchanged messages with someone else, so maybe the problem isn't FB or Messenger but a friend who isn't checking messages. Do I need this app since, except for this recent incident, Messenger seems to be working normally? What about the unsupported browser notifications?