I have an issue with a remote computer running Windows 10. As soon as the screen switches off, that PC disappears from the list. Running a fresh install of the latest 32 bit version of Win10 (10.0.15063 Build 15063) and TeamViewer v12.0.81460
If TeamViewer is not in session to the remote machine (an Asus T100A) and that remote machine switches its screen off after its pre-defined period as set in the Power & sleep settings of Windows 10, it will drop out of the available list of remote computers straight away and not come back until the screen wakes up. When I look at that machine locally and wake the screen up, I notice that the TeamViewer icon in the taskbar is showing a red exclamation mark but then quickly disappears and it comes back up as available.
I have gone through all Windows 10's power settings and switched every conceivable sleep or power saving feature off, to no effect. The network is still active (not gone to sleep) because I can Skype into it even though that PC has disappeared according to TeamViewer.
Interesting though that nudging the remote with a Skype call will switch the screen on and therefore, revive it. Somewhat of an undocumented security feature! Or at least, a workaround to the above problem. But even then, only the Store app version (metro?) of Skype will do this.
The other workaround is to set the screen to never switch off. What is surprising though is if I close the lid to the remote PC, which also switches the screen off, the problem does not come into effect and the computer, despite having its lid shut and screen off, remains online to TeamViewer. BUT if I set the screen to switch off in Power & sleep settings AND close the lid, after the pre-determined time the PC will drop off the list.
I have the exact same problem. TeamViewer was working fine for many months but then a couple of weeks ago the remote computer started disappearing from the My Computers list as soon as the PC display shuts off. The remote PC is set to never sleep, so that's not the issue. I'm using TV version 12.0.82216 on both PCs.
Interestingly, Chrome Remote Desktop still recognizes the PC when the screen is off, even though TeamViewer does not. So the problem seems to be something in TeamViewer that changed.
Since I prefer TeamViewer over Chrome RD, the workaround I use is to access the remote PC using Chrome RD and, once the screen is on and TeamViewer can see the PC, I use TeamViewer. It's inelegant and kludgy but works.
It would be nice if someone from TeamViewer actually monitored and responded to these posts!
My use case is different in that I use TeamViewer mostly to access my home PCs when I'm at work. I'm a user and not an IT guy I'm afraid. I haven't had the second problem you describe. The only other issue I've had is losing the remote PC altogether in TeamViewer becuase the remote PC lost power and went to sleep. I don't use the Wake on LAN option but might try that.
What I find strange is that it used to work fine and suddenly stopped when nothing changed on the remote PC, and that it works fine with Chrome Remote Desktop, which is free. I'm not sure if the problem started after a TeamViewer upgrade or not.