Shutting down of Raspberry pi
The red LED being on after shutdown is normal.
When you shut down the green LED will start blinking very fast, which means it is writing to SD card. When the screen has disappeared and it is safe you will see a fast regular blinking pattern on the green LED: ten times on/off and then all the LEDs but the red one will go out. Then it is safe to remove the power.

No, the Pi does NOT turn off its power like .e.g. a PC. 

By the way: 


The Raspberry-Pi stack exchange is the better place to ask this: https://raspberrypi.stackexchange.com/questions/42929/how-to-safely-switch-off-the-raspberry-pi/42945
Since a few years it is a lot more difficult to mess up the SD-card. Not impossible but a lot more then it was in the begin period.
All this you can find on the internet in many places.