This problem can be solved by preschool children in five to ten minutes, by programmers in an hour and by people with higher education…well, check it yourself.
An arrow. It has feathers and its nest (a quiver) is usually carried an archer’s back. The arrow flies when released, but will always land somewhere to rest.
I make memories that last a lifetime,
Without me you’ll be miserable,
If you miss me you can gourge and it’s okay.
Some say I am naturally split in two.
Sleep. It’s the time your mind organizes memories and knowledge. Sleep deprivation is miserable (as you well know if you’ve experienced it). If you miss a few nights of sleep, you can sleep more to catch up with no problems. And some believe that segmented sleep, where you sleep, wake for an hour, then go back to sleep, was the dominant form of sleep before the Industrial Revolution.
Because Oct 31 represents the octal (base 8) number 31, which, when converted to decimal, is 25. Dec 25 is short for Decimal 25, thus the two are equal.