Mobile devices in general are good at going into a very low power mode when they aren’t in use (CPU and other apps go into very low power mode). This is a function of Android (although I am not sure we are using exactly the same terms as Android).
In general, “deep sleep mode” and “power save mode” refer to when Android is trying to keep the CPU in an ultra low power mode.
Yes. You can get some battery monitoring apps on Android and many of them will point out how long your device is in each power state (like Deep Sleep).
The two power save modes mentioned are actually affecting the agent itself. These modes will suspend the client from running when the environment is in a stable state, times which vary depending on battery and connectivity. The Deep Sleep mode is a mode where the client will not respond to many external deployment events and goes into the deepest sleep to save the most power - not to be used without full understanding of the limitations it brings.