1. There is no such SEPARATE thing called "division" as you've learned it(contrary to school teaching).
What you DO have, is the following:
Given any non-zero real number "a", there exists another number, that we have given the name "(1/a)", that has the property when multiplied with "a" yields the number 1, that is: a*(1/a)=1.
We can prove that for any particular number "a", the number "(1/a)" is UNIQUE, so the identity a*(1/a)=1 can be said to DEFINE (1/a) (just like we DEFINE "2" to be the number so that 1+1=2, really)
Whenever we multiply (1/a) with some other number, say b, that is we form the product b*(1/a), we find this notation so cumbersome so that in short hand, we introduce the notation b/a=b*(1/a).
Now, we can prove that whenever we multiply a number with 0, then we get 0, so for any "a", we have 0/a=0*(1/a)=0, providing the answer to your first question.
The answer to your second question is also contained here, because if 0*anything=0, then, since 1 isn't 0, then there CAN'T exist a number (1/0) having the property 0*(1/0)=1 (remember I required that a had to be non-zero earlier?)
This essentially answers your questions.
2. What you compute at school, and calling the "operation of division" is quite simply how to find the decimal representation of the fraction b/a.
This is entirely analogous to that what you call "operation of addition", where you merely are requested to find the decimal (or, rather denary) representation of, say, the number 23+49