Keeping the earbuds in the charging case is safe because they're completely turned off that way. What is NOT safe is keeping them out in an environment with high temperature because the heat will degrade the batteries faster and the earbuds will be left on.
Yes, place them back into their case so that way they can recharge. I don't believe they auto shut off because they have a feature to find them if you lose an earbud and that feature would not work if the earbud was off.
Keeping the earbuds in the charging case is safe because they're completely turned off that way. What is NOT safe is keeping them out in an environment with high temperature because the heat will degrade the batteries faster and the earbuds will be left on.
