Basic uderstanding of Echo devices and zigbee connections

[marklyn59]

I thought I understood the connection between my Echo studio and the zigbee devices it controls but I think I need a better understanding.
I have an Echo Studio and two echo dots.
I seek to understand how my Echo Studio communicates with my various lights and plugs, all of which are Amazon branded and supposed to work directly with my ES via zigbee directly.
When I am home and ask Alexa to turn on a Amazon Hue light, does it need the internet to do this?
When I'm away from home and use the app to turn on a light, what is the command pathway for this to take place?
Lastly, what reasons would I get a "server unresponsive" when using the app to turn on a light when I am 100% sure my internet is running fine and Alexa says she's connected to the internet?
If there is a web page that shows these connections visually, I'd appreciate a link to that.