Why am I not getting notifications from the App Store about available updates? I have an iPhone 7 (32GB) with iOS 10.3.3 and it seems like I'm not getting any notifications like "This App was updated" or "Some Updates are available" or any kind of Badges of available updates on App Store.

I have turned on All Notifications for App Store in Settings including Banners and Badges. I had my iPhone reset like 3-4 times (including Restore via iTunes) and it just didn't work. I have to manually go into the App Store every time and push it down to check for any available updates. Why can't it just happen automatically. I have seen updates badges on many other iPhones.
You can enable automatic updates in Settings → iTunes &amp; App Stores: switch on Updates under Automatic Downloads.

When an app update is available, the App Store badge will be shown and the app downloaded at a later date. You won't get notifications for updates, pre- nor post-update. A blue dot will be shown next to app names for apps that have been updated.

Once settings are correct, restart your phone and consider resetting your network settings. 

Next, check if an outage is happening: https://www.apple.com/support/systemstatus/

Last step, use the contact us link at the bottom of Apple online system status if you have an issue that’s not reported.