So, after the sketch is uploaded, and I want to turn the Arduino off while it is running a sketch, all I have to do is unplug it. Should be safe as long as nothing is uploading. But I wonder, if I use serial communication between the Arduino and my Computer in a sketch, does it harm either if I simply unplug it?
if the sketch is uploaded, it wont hurt it to unplug. Just give it another source of power and it will run its sketch.
You can unplug it, it does not harm. If your arduino does not need power, eg. for writing EEPROM or FLASH, then no danger there. (in case it was writing EEPROM/FLASH it could write it faulty but usually even that is not a problem either)
Yes, you can unplug it directly whether if serial communication is active or not if you dont need power supply anu more using usb cable