Hi Sun,
Welcome to Microsoft Q&A forum.
This is Joy from Microsoft Device & Deployment support team, I will try to answer your issue about power management.
What's your device model? Is it a virtual machine or physical machine?
I noticed you said powercfg /a show s1,s2,s3 all not available on your deivce. So rundll32.exe powrprof.dll,SetSuspendState 0,1,0 command line not work on your device is an expect behavior as hardware haven't related sleep mode to go into.
I noticed you could "sleep by Win-power-sleep item correctly". Please go to Event Viewer\Windows log\system and check if any event ID 1\42 recorded.
Event ID 42 in the System log from source Kernel-Power is the sleeping event. Event ID 1 in the System log is for waking. We could check event reason for it.
For the second issue, we could configure a wake up scheduler, please refer to How to Make Your PC Wake From Sleep Automatically
For the last issue, we could try Take-Screenshot . As we are not familiar with command line or script, so I would recommend to create a new thread and add tag with "Windows-Server-PowerShell" for consulting command or script issue.
If my information is useful for you, please accept it as answer.
Bests,
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