With a pulley of 5 strings, why is each string experiencing 10 N of force and not 2 N apiece (when you pull on the string with 10 N of force)?

When you pull on the string with a 10 N force, you create 10 N of tension in that string. If there is less tension anywhere in the string, then that portion of the string will accelerate toward the side with more tension. That's why the tension in each string of a multiple pulley is 10 N when you pull on its loose end with a force of 10 N. The 5 strings are really just parts of the same string and that string has to have 10 N of tension in it.

What is the difference between a multiple pulley system in which the string you pull on comes down from the top pulley and the one in which the string you pull on comes up from the bottom pulley?

When the string you pull on comes down from the top pulley, it doesn't exert its tension on the thing being lifted so it doesn't count when add up the strings. But when the string you pull on comes up from the bottom pulley, that string is also helping to lift the object. That string does count. Thus if the multiple pulley has 5 segments going up and down between the two pulleys and one more segment going up to your hand, the total number of segments lifting the object is 6 and that object experiences an upward force equal to 6 times the tension in the string.