I'm running on Ubuntu 16.04 and I have the same issue on mine with 1.5.2, however I also had it with 1.5.0 and the nightly builds I tried.
While I can't answer your question 1 as I usually just grab the whole Minecraft folder to get settings and whitelist etc I found that Stop/Start server also work in tasks (though haven't found any others I can successfully scheduled after trying all of the send message to players or send console input to server). It'd be nice if those worked so I could warn people before backups. But since scheduled backup task won't work either I have to make do until that task works.
Warning: Use my workaround or scripts at your own risk, I'm not responsible for any issues that might arise.
To solve my own backup issues, I set the "Stop minecraft server" task for 4AM (seems to work).
I then made a cronjob for 4:02 AM:
AMP@localhost:/backup/minecraft$ crontab -l
# COMMENTS REDACTED TO SAVE POST SPACE
2 4 * * * /bin/bash /home/AMP/backup_minecraft.sh
To run a simple backup bash script:
# Backup Minecraft folder from MCMA instance
tar -cvzf $backup_name /home/AMP/.ampdata/instances/MCMA/Minecraft
And then at 4:05 I set another task in AMP to start the server (since my backups are small 3 minutes is currently plenty of time for my situation)
Since the server is only vanilla and small number of players none of this needs much time. Tonight will be the first test run but so far I've used different time settings to do a daytime test with success of the method overall.