I'd like to figure out how to hide the icons of the status bar that would appear in the left. Is there a way to control that? Some apps such as Weather.com don't have the option to hide that icon from the status bar.
Interestingly, you don't need an app to hide those notifications. You can go to the App Info screen for an app and uncheck the "Show notifications" option. There are a few ways to get to the App Info page for an app.
Let's also take a moment to learn the reason why apps (like weather apps, etc) don't have an option inside the app itself to do this!
If an app requires its notification to be displayed 100% of the time, then Android's Garbage Collector/Process Manager is not able to automatically kill it. If, for example, The Weather Channel's app allowed you to disable the notification/icon, then using that option would possibly allow Android to kill the background process (without knowing that it's important) and it would stop getting weather updates.
Services that use Push Notifications (Like Hangouts, Email, lots of chat apps, etc) don't need to be constantly running in the background to function. However, apps that need to do more background processing or need richer data (I guess weather apps?) need more than just push notifications. That's why this is a thing.
Fortunately, disabling notifications inside the App Info screen does not affect the app's other functionality. The app is still requesting the notification to be shown (Therefore protecting itself from getting killed) but the SystemUI App is just configured not to show it.
Try Notifications Off.