That chat bubble is there because you granted Messenger the permission "draw over other apps". In simple terms, call it overlay. The qualified name of the permission is
Allows an app to create windows using the type
TYPE_SYSTEM_ALERT, shown on top of all other apps. Very few apps should use this permission; these windows are intended for system-level interaction with the user.
I don't think chat bubbles of Messenger, in any capacity, are meant for system-level interaction with its user, so you can assume that developers of that app and any such app chose to abuse the system's feature for their own interests.
Furthermore, you definitely are not the first person who thought about exploitation of the overlay feature of Android.
ArsTechnica: New Android ransomware locks out victims by changing lock screen PIN
Dubbed Android/Lockerpin.A, the app ... overlays a bogus patch installation window on top of an activation notice. When targets click on the continue button, they really grant the malicious app elevated rights that allow it to make changes to the Android settings. From there, Lockerpin sets or resets the PIN that unlocks the screen lock, effectively requiring users to perform a factory reset to regain control over the device.
Once the continue button is pressed, the app will acquire administrator rights [becomes a Device Administrator]. From there it will change the PIN and periodically continue to overlay a fake window in an attempt to hold on to the elevated privileges. It's the first known Android lock-screen ransomware to set a phone's PIN lock. Because it requires non-paying victims to factory reset their phones, it causes them to lose all of their data.
Just so you know, Android understands the potential damage overlay can cause to a device, hence, when an overlay is actively drawn on screen and user attempts to sideload an app, the install button simply fails to work. See Why can't I press the Install button when installing applications from unknown sources?
If only Android extends this feature to activation/deactivation of device administrators, that would be very nice of them.
In case, you want to know which other app has the same permission, you can use my answer on Determine which app is drawing on top of other apps?
The notifications you mentioned, intrusive they may be, are called Heads-up notifications. Until Android 5.1, users didn't have an official way to get rid of them without losing all notifications from an app.
Android Design: Heads-up Notification
When a high-priority notification arrives ..., it is presented to users for a short period of time with an expanded layout exposing possible actions.
After this period of time, the notification retreats to the notification shade. If a notification's priority is flagged as High, Max, or full-screen, it gets a heads-up notification.
As you may have guessed, Heads-up notifications are not the default type of notification a system should be displaying to a user and a developer have to explicitly opt for it in the app's code. It's just that some developers opine that the things their app do are of the ultimate priority in the life of the user, let alone the system, so they often choose to show every notification as a Heads-up notification.
That said, heads-up notifications were introduced only in Lollipop while your device runs Android KitKat, so definitely, as Andrew T. put it in comments, it must be Facebook's native code which is both creating and displaying those pseudo-Lollipop notifications at the top and above a window.
The reason you can't interact with system bar when that pseudo-Lollipop notification is active, is most likely the use of blocking overlays to display a view which is perceived as a notification by the user. Aaron's answer has explained it with a good working example app.
I searched web between the range Oct 2013 to Oct 2014 (the developmental age of Android 4.4.x) and came to know that there were apps available in Play Store primarily meant to create those pseudo-Lollipop notifications. One such app is Floatifications and the other one I found was Metro Notifications Free. Both uses the same blocking overlay to create those pseudo-Lollipop notifications.
I'm surprised you didn't find toasts pesky enough.