What is $0\div0\cdot0$?

whatever happens when we divide that first 0 by 0 should be reversed when we multiply it by 0 again

I think this sentence reflects a fundamental misunderstanding of what it means when people say "$0 \div 0$ is undefined". It is not helpful to think of "undefined" as the name for an exotic, mysterious number-like object of some sort that is produced by the action of dividing $0$ by $0$. It is more helpful to think of "undefined" as a signal that you are trying to ask a question that has no answer.

When you write "$0 \div 0$" you are essentially asking the question: What is the unique number $r$ with the property that $0 \cdot r = 0$? That question doesn't have an answer at all, so $0 \div 0$ is a meaningless expression.

Now it is certainly possible to decide to introduce a new symbol into your mathematical world -- let's use 😱 (the emoji for "face screaming in fear") -- and declare that whenever we encounter the expression $0 \div 0$ we can replace it with the new symbol 😱 . Then your question is, "What is 😱 times 0?" And the answer to that question is, "We can't answer, because we don't know what 'times' means in this context." You may as well ask "What is pickle times lumberjack?" The question doesn't have an answer because "times" doesn't have a pre-existing meaning when one or both of the things being "multiplied" aren't numbers.

If that result is unsatisfying, you can try to extend your number system by defining a new rule: "From now on, I declare that 😱 times any real number is 😱 ." Or perhaps "From now on, I declare that zero times anything -- whether or not the anything is a number -- will always be zero." Or you can say "From now on, I declare that 😱 times zero is $1$." You can pick such a rule if you want to, but having made such a choice, there will be consequences. In particular it might turn out that in your new extended number system, you have to abandon the associative or commutative properties; alternatively, if you insist on keeping those properties, it might turn out that all numbers can be proven to be equal to each other, so the entire system collapses.

Ultimately, all mathematical definitions are conventions, and all conventions are local. So you can decide to define the expression $0 \div 0 \cdot 0$ any way you want to, if you really want to. But while you can define the expression any way you want, you have to live with the consequences, and the sad truth is that there really isn't a way to make the expression $0 \div 0 \cdot 0$ meaningful in a way to make it compatible with the other operations and properties of arithmetic.

[W]hatever happens when we divide that first $0$ by $0$ should be reversed when we multiply it by $0$ again[.]

Is it not equally "true" that if you multiply something by $x$, you can reverse whatever happens by dividing by $x$?

For example, if this sort of thing works, we should have $(3 \times 0) \div 0 = 3$ and $(17 \times 0) \div 0 = 17$. (The parentheses are there just to emphasize that we take a number, multiply it by zero, and then divide it by zero.)

But $3 \times 0 = 0$ and $17 \times 0 = 0$, so now we are saying that $0 \div 0 = 3$ and also $0 \div 0 = 17$.

This runs afoul of other commonsense notions, such as the notion that two things each equal to a third thing are equal, the notion that the result of a binary operation on two numbers should be another number, and the notion that there is one number named $3$ and that $3$ is always that same number and not equal to $17$.

You can probably think of some set of rules to avoid this mess, but in the end they will come down to saying that you can't compute $(17 \times 0) \div 0$ and have it come out to the "commonsense" answer. How about the rule "do not divide by zero"? It's very simple, and it solves the problem.

Just because one function is the inverse of another for infinitely many cases doesn't automatically mean that it's the inverse for all cases.

Here's a simpler example for you to consider: the Collatz problem. The Collatz function is defined thus: $f(n) = 3n + 1$ if $n$ is an odd integer, $f(n) = \frac{n}{2}$ if $n$ is an even integer. We can also define a doubling function $d(f(n)) = 2f(n)$ and a "thirding" function $c(f(n)) = \frac{n - 1}{3}$. If $f(n) \equiv 4 \pmod 6$, which is the inverse function of $f(n)$? The doubling function? Or the thirding function? If we're not keeping track of the iterations, then we just have no way of knowing.

Suppose $n = 7$ and $f(n) = 22$. If we put $f(n)$ through the doubling function we're not going to reverse $f(n)$ because the Collatz function was fed $7$, not $44$.

And so it is with the "zero function" $z(x) = 0x$. Whatever number $x$ is, $z(x) = 0$. If we only know what $z(x)$ is, we have no way of knowing what $x$ is, it could be any number at all.

But this zero function assumes you're giving it a number. Turns out $\frac{0}{0}$ is actually not a number. As others have already indicated, you can really wear down your brain constructing an argument that seems to define division by $0$ only to find it demolished by a slightly different argument. One of the arguments presented so far demonstrates three possible values for $\frac{0}{0}$. When you learn about complex numbers, you will see the possibilities become infinite, but no more valid.

So $\frac{0}{0}$ is undefined and invalid, and trying to multiply it by $0$ or any other number is pointless.

Think about what it means for $x \div x$ when $x$ is any nonzero real, imaginary or complex number. We have $x \div x = 1$. This suggests that $0 \div 0 = 1$. But what about $2x \div x = 2$? We can then argue that $0 \div 0 = 2$. Whatever nonzero number we want $0 \div 0$ to be, we can argue for it.

Let's say that we agree that $x \div 0 = 0$, as if somehow that repairs the discontinuity in a $c \div y$ graph, where $c$ is some nonzero number. What does that gain us? Not much. If for nonzero $x$ we have $x \div y = w$, then $x = wy$, but this is false if $y = 0$ because $0w = 0$. Multiplication still can't be relied upon as a calculation history look-up function.

Much has been made of $\sqrt x$ being not technically a function. If $x = y^2$, then there are two possibilities for $y$. But that's only two possibilities, and one can be obtained from the other with multiplication by $-1$. If we define $x \div 0$ to be some specific number, then there are infinitely many possibilities for solutions to $x \div 0 = w$.

Another problem with defining $x \div 0$ to equal some specific number is that then we have to re-examine what $0x$ is. But since $0x = 0$ has worked for us so well for centuries, we have to ask ourselves if defining $x \div 0$ is really so worthwhile that we may possibly have to redefine $0x$.

If we agree that $0x = 0$, then we just have to accept that multiplication might not be a reversal of division when zeroes are involved. So even if we were to agree upon $x \div 0 = 0$ (which we don't) we'd have $0 \div 0 \times 0 = 0$ but we'd also have $1 \div 0 \times 0 = 0$, $2 \div 0 \times 0 = 0$, etc. ad infinitum (not to mention all the rational and irrational numbers).

The only way that multiplication can be a consistently reliable reversal of division is for it to be defined that way.

To use computer programming parlance, $0 \div 0$ is "not a number" and throws some kind of exception. If instead of dealing with the exception we try to multiply that "not a number" by $0$ then we get another instance of "not a number" and another exception.

What you are saying sounds reasonable enough, and so it is worthy of serious consideration.

You're saying that $a \div b \times b = a$, which is in fact true for all nonzero values of $a$ and $b$.

Then, if $a = b = 0$, we have $0 \div 0 \times 0 = 0$. That looks correct. So far so good.

What if $a = 1$ and $b = 0$? Then $1 \div 0 \times 0 = 1$. Whoa, that doesn't look right. This would mean there is a solution to $x \times 0 = 1$. But we can prove that $x \times 0 = 0$ no matter what number $x$ is.

Considering your idea, we have arrived at a contradiction. (This is not the only contradiction possible, though, there are others).