‘Idle’ from a Sonos perspective is defined as audio muted or paused on all players in the household for at least 3 minutes, and no audio signal is being sent to a line-in connection or home theater player (in most cases, this means that the TV is off).

All Sonos components take 3 minutes to go into idle mode, unless part of a Sonos home theater surround setup. The 3 minutes until idle can be initiated on a surround setup when audio is paused. If pause or mute is not initiated, idle power mode takes 13 minutes for all devices part of a surround setup after the audio signal ceases from the TV.

For your TV to power off using Alexa, you need the Sonos Skill installed in the Amazon Alexa App and the Sonos TV control device in your list of devices. Sometimes, renaming the control device in the Alexa App to "TV" may work in some instances, but note the TV itself needs to support the ‘power off’ command as part of its own HDMI-CEC control function too .. It maybe the case that the only way to power off the TV is by using its assigned ‘remote’ control only and this therefore maybe ‘by design’, but perhaps check your make/model with the TV manufacturers Support desk to see if the power off feature is supported using the CEC protocol.

Enabling Alexa will not prevent the Beam from going into idle mode. 

There is no way to tell it’s in idle mode other than knowing that all Sonos devices take 3 minutes to go into idle mode after not receiving an audio signal for 3 minutes.

From Sonos:

All Sonos components take 3 minutes to go into idle mode, unless part of a Sonos home theater surround setup. The 3 minutes until idle can be initiated on a surround setup when audio is paused. If pause or mute is not initiated, idle power mode takes 13 minutes for all devices part of a surround setup after the audio signal ceases from the TV.

Here is an article about what you can ask Alexa to do with Sonos:

https://support.sonos.com/s/article/3514?language=en_US