Are you denying this as somebody with strong knowledge of mathematics?

(I need to know what prior I should assign to this conceptualization being wrong. I got it from a mathematics instructor, quite possibly the best I ever had, in his explanation on why canceling out denominators doesn't fix discontinuities.)

ETA: The problem he was demonstrating it with focused more on the error of -adding- information than removing it, but he did show us how information could be deleted from an equation by inappropriately multiplying by or dividing by zero, showing how discontinuities could be removed or introduced. He also demonstrated a really weird function involving a square root which had two solutions, one of which introduced a discontinuity, one of which didn't.

Qiaochu_Yuan already answered your question, but because he was pretty technical with his answer, I thought I should try to simplify the point here a bit. The problem with division by zero is that division is essentially defined through multiplication and existence of certain inverse elements. It's an axiom in itself in group theory that there are inverse elements, that is, for each a, there is x such that a*x = 1. Our notation for x here would be 1/a, and it's easy to see why a * 1/a = 1. Division is defined by these inverse elements: a/b is calculated by a * (1/b), where (1/b) is the inverse of b.

But, if you have both multiplication and addition, there is one interesting thing. If we assume addition is the group operation for all numbers(and we use "0" to signify additive neutral element you get from adding together an element and its additive inverse, that is, "a + (-a) = 0"), and we want multiplication to work the way we like it to work(so that a(x + y) = (ax) + (a*y), that is, distributivity hold, something interesting happens.

Now, neutral element 0 is such that x + 0 = x, this is by definition of neutral element. Now watch the magic happen: 0x = (0 + 0)x
= 0x + 0x So 0x = 0x + 0x.

We subtract 0x from both sides, leaving us with 0x = 0.

Doesn't matter what you are multiplying 0 with, you always end up with zero. So, assuming 1 and 0 are not the same number(in zero ring, that's the case, also, 0 = 1 is the only number in the entire zero ring), you can't get a number such that 0*x = 1. Lacking inverse elements, there's no obvious way to define what it would mean to divide by zero. There are special situations where there is a natural way to interpret what it means to divide by zero, in which cases, go for it. However, it's separate from the division defined for other numbers.

And, if you end up dividing by zero because you somewhere assumed that there actually was such a number x that 0*x = 1, well, that's just your own clumsiness.

Also, you can prove 1=2 if you multiply both sides by zero. 1 = 2. Proof: 10 = 20 => 0 = 0. Division and multiplication work in opposite directions, multiplication gets you from not equals to equals, division gets you from equals to not equals.

You can do the same thing in any system of logic.

In more advanced mathematics you're required to keep track of values you've canceled out; the given equation remains invalid even though the cancelled value has disappeared. The cancellation isn't real; it's a notational convenience which unfortunately is promulgated as a real operation in mathematics classes. All those cancelled-out values are in fact still there. That's (one of) the mistakes performed in the proof you reference.

Not necessarily true. A good rule for introductory math students, but some advanced math requires dividing by zero. (As mentioned, that's what a derivative is, a division by zero.)

Limits are a way of getting information out of a division by zero, which is why derivatives involve taking the limit.

Division by zero is kind of like the square root of a negative number (something introductory mathematics coursework also tells you not to do). It's not an invalid operation, it's just an operation you have to be aware of the ramifications of. (If it seems like zero has unusual behavior, well, the same is true of negative numbers with respect to zero and positive numbers, and again the same is true of positive numbers with respect to zero and negative numbers.)