Do users get charged for sending private messages?

No, private messages to any model does not reduce the token balance, or even require for there to be a non-zero balance.

Can the model and user send media to each other over messaging?

Yes, No, currently this is not a feature that’s present in the base product. If this is something you want to have on your site, please let us know for customization.

Do the user messages expire if the model is not online?

No, they don’t. Models can check and respond to the messages when they login later.