But mixed results so far. When it was running, my mobile device and laptop could connect to the server, but neither one of my smart TVs could. Not sure what is going on there, since they both have the Plex app,
Your mobile device and laptop prove the server is running. Maybe you have a network issue? Can the smart TV ping the NAS. (Or can the NAS ping the TV, if that is easier).
Is it safe to install/update the newer version?
It's safe as in it won't kill your NAS. Don't know if the new package will run. On the NSA300 series the 1.x server doesn't run, due to wrong libraries. But the NAS500 is a totally different box, so I don't know. Worst thing that can happen is that you'll have to reinstall the older version, I guess.
Also, when the NAS goes to sleep and after I wake it up, the Plex Media Server is not running. I manually re-booted the NAS and it started back up automatically, but is there a way to have it startup/run when the NAS wakes up?
What do you mean with 'goes to sleep'? Disk spindown or box switches off?
When the startscript has the executable bit set it will be called on boot. There is no way to start the server when the disk awakes, but the server is not supposed to stop when the disk spins down.