Does Windows Phone support app widgets like Android phones do?
No and there's no need.
One of the key features of the Windows Phone platform are Live Tiles, which provide a snapshot of information on your Start Screen. We have Live Tiles that will deliver weather information, appointments, TV shows, and even gaming progress.
Personally, I like how I can glance at the weather and appointment Tiles to see what the current temperature is or how long I have until the next meeting. Other than that, I generally use the Live Tiles as a shortcut to the many apps I have pinned to my Start Screen.
I cannot think why would anyone want widgets when we have live tiles