Apparently, the "Events" tab on the app (the specific child app, not the one/big parent app) does show a history of notifications that child app generated. The "Events" tab is usually not a useful spot to look for apps, so I had to check this myself to see. Not sure if that's what you meant with "logs," but if not, that's one spot you could check to see whether the app thinks it sent its notification.
Often when people see problems with this, it's really a problem with the mobile app. Try going to your "Devices" list, then find your mobile app device. Click into the page for that device, and try sending a test notification to your phone. Just type some text in the "Device Notification" box and click the button to send it. If this doesn't work, there's a problem with the app. Logging out and back in to the app may help, though it's usually pretty easy to just delete/uninstall and reinstall the app (not much data stored in it).
If you did get your test notification, you shouldn't need to bother with the above, so it likely is a problem with the Notification child app, assuming you've selected the correct notification device. What's likely the case is that your devices aren't reporting status to Hubitat, so the Notifications app never "knows" it's suppsoed to fire. You mentioned locks and lights. Both of these can be tricky. When you manipulate your lock (say, manually lock it), do you see this change reflected in Hubitat? Also, what kind of lights do you have? Newer smart switches/dimmers should be OK, but I'd still verify that you see their on/off state change in Hubitat when you manipulate the device physically (or even digitially, though that's not been historically problematic). If it's an older Z-Wave switch/dimmer, you may need polling to get status updates, though I'd recommend replacing the device. Or if you're using smart bulbs, what kind, and do you have any other Zigbee devices (assuming your bulbs are Zigbee) on this network? Many Zigbee bulbs do not play well on mixed Zigbee networks.
For IFTTT, I don't think Hubitat's native integration supports arbitrary notifications (you can get notified when a specific IFTTT applet runs). There is a community driver that can send arbitrary notifications, but it hasn't been touched in over a year and I wouldn't be surprised if nobody is using it anymore since the release of the mobile app.
