Sorry for the late reply. And you don't sound stupid, don't worry! So, there are two sync sections for Facebook, one in your Facebook Account Settings, which lets you sync calendar, photos, and contacts and since you don't want these, you should have those turned off. However, there's an Android Master Sync Switch, and this basically lets you get push notifications and sync data for apps (if you turned, say, calendar sync for Facebook off in Facebook's own settings, then you're still not syncing the app no matter what's the state of this switch, but if you chose to sync something and this master switch is off, then nothing is getting synced). But if this switch was off, then you wouldn't, in theory, get the messages on mobile data or WiFi.
Now, there MIGHT be a problem with your carrier. I know it's a long shot, but I've had issues with push notifications while on certain WiFi networks due to firewalls and proxies (for instance, I can't get push notifications for my e-mail while at the office's WiFi but as soon as I connect to mobile data or another less-restricted WiFi network, all notifications come pouring through. Sounds like what you're experiencing, but in my case, I know it's a router/firewall issue.