Answer to last puzzle: Yes, the arrow will meet its mark, because it is possible for an infinite summation of numbers to equal a finite sum. In other words, we can add the "half distances" infinitely many times, and it can produce a finite "total distance".
Although it has its foundation backed by mathematics, I don't like this answer! Some of the answers I received in the comments were much more intriguing and founded in physics roots or other more tangible properties:
"The number gets so small that it is good enough to be called touching. besides. Due to the fact that atoms exert a force on other atoms nothing ever really touches anything. we only experience the forces of the atoms pushing on one another." -vashrave
"I would say no as even if it is halved to infinity, it would eventually be traveling a distance of Planck length(1.163*10^-35).
IE: Much smaller than a neutrino which is so tiny that it can pass through matter almost undisturbed. Strings(1D) in String theory exist at this level.
Any length shorter than Planck length makes no sense given the laws of physics and quantum mechanics that are agreed upon by the scientific community today." -Mike
"I thought its a simple limit of a geometric series. Mathematically, the formula for the series of each halving of the distance will be 5/(2^n) from n=0 to n=infinity. As |r| < 1 the series diverges and the entire distance traveled is the sum of this series. The formula for the entire distance traveled is a/(1-r) where a=5 and r=0.5 (from 5/(2^n)) So the total distance traveled is 5/(1-0.5)=10. Sorry if this is hard to follow, it made more sense in my brain." -TheGeneral
TheGeneral's was probably the closest to my mathematical explanation. In response to Jacob who said, "Good rule If you have a roommate; Take no more than half the sugar.. that way there will ALWAYS be some left^^" This isn't exactly true, because eventually you'll reach the last grain which you can't half (not easily, anyway). In other words, there's a fundamental unit; time and space, on the other hand (as the question is composed of) has no fundamental unit so it can be infinitely halved. Nice try though, Jacob =).
You are given four large wooden letters on a table in front of you. They are the lower case "a", the lowercase "n", the lowercase "t", and the lowercase "h". Arrange these letters to make a hexagon. You may not use a curved side as a straight side.