Rose says her Facebook has been hacked. She keeps changing it, but she's still having issues with her Facebook account posting things and tagging everyone she knows. Leo says that in the past, Facebook has had security issues with accounts being hacked, but as far as he knows, they've fixed all the exploits. So here's a few things Rose can try:
She should turn on 2nd Factor Authentication. Then she should go into her app settings and disconnect all the apps listed. That's likely where the breach is coming from. Also, in the app settings, she can look to see where she is logged into Facebook. It could also be a browser plugin, so she should remove those. Something is likely on the laptop that is using Facebook to spam. She should also clear her browser or reset it. Uninstall Java and Flash as well. Leo also recommends a program called Little Snitch. It'll watch outbound traffic and let her know if something is running on her computer and sending outbound traffic.
A long shot is that her computer could be hacked via remote access. Rose could try disconnecting her computer from the internet for a few days and see if it persists. If it doesn't, then she'll know her computer has been hacked. But it's unlikely because Rose has a Mac.
Leo also says that if she uses the https URL, then it should put her in a secure mode when she goes on Facebook. She can click on the padlock and check the certificate. If it's not from Facebook, then there's a man in the middle attack going on using something like Avast.