Q:

Is it correct that $\frac{1}{0}=\frac{1}{-0}$ and if it is, why is $\frac{1}{0} \neq 0$?

This is a genuine question, I am not trying to convince anyone. But I'm sure hundreds of people already considered this, so if you can point out where I'm wrong, it would be much appreciated.
If we forget about limits and infinitesimals, then $0 = a - a$, where $a \in R$
Then $0=-0$
$\frac{1}{0}=\frac{1}{-0}$
$\frac{1}{0}=-\frac{1}{0}$
$\frac{1}{0}+\frac{1}{0}=0$
$2 \frac{1}{0}=0$
$\frac{1}{0}=0$
It also means that $\frac{0}{0}=0$
Then the usual reasoning that $\frac{0}{0}$ can't be determined because for example $2*0=0$ and also $3*0=0$ doesn't hold. We can divide both sides by $0$, but we will get only the trivial $0=0$.
$\frac{1}{0}=0$ may seem counterintuitive, but since the limits for 1/x approach $\infty$ from one side and $-\infty$ from other side, it makes sense that the value for the exact zero will be exactly in the middle.
I can offer one counterpoint for my reasoning. The 'proof' I provided is incorrect if and only if 1/0 does not belong in the set of real numbers. Then the addition of this value may not be commutative. But I'm not sure that there is any proof that 1/0 is not in R.

Edit:
Thank you all for your answers, and I apologize for trying to argue with some of you in the comments. I am not trying to start a discussion.
I consider my question answered now, in fact, I have already answered it myself in the last paragraph.

A:

Both expressions
$$\frac 10 \quad\text{ and }\quad \frac 1{-0}$$
do not make sense with the usual rules of the real numbers. 
(If they did, then $0$ would have a multiplicative inverse. Namely, there would exist a real number $x$ such that $0\cdot x = 1$. But it is straight forward to show there is no such number $x$, since $0\cdot a = 0$ for all real numbers $a$.)
As such we cannot perform algebraic operations with those expressions and reach valid conclusions.
To put it more bluntly: we cannot start with a nonsense statement
$$\text{"Meaningless expression"} = \text{"Meaningless expression"}, $$
which is not even a false statement, and logically conclude a true statement.

A:

What division really means is the inverse of multiplication. So 1/0 means the number you multiply by zero to get one. However, it is a property that x*0 = 0 for all x. This is the reason we say 1/0 is undefined. 
This is all a consequence of the interplay of operations (addition and multiplication) on the space of real numbers. Have you studied rings?

