Note: This post can't be considered as a personal theory. These are the arguments which I figured out by reasoning and I just want to know it's validity.
Here I present multiplication and division operations of $0$ and $\infty$ by pure reasoning. $$\dfrac{x}{0}=\infty$$ (where $0$ is positive infinitesimal) $$AND$$ $$\dfrac{x}{0}=-\infty$$(where $0$ is negative infinitesimal)
If we divide $x$ by a small no., we get a big no. $$\dfrac{x}{small}=big$$
If we keep on dividing $x$ by a small no. and then by another small no. and so on, we get bigger and bigger no. $$\dfrac{x}{small\times small\times.....}=big\times big\times.....$$
Similarly if we divide $x$ by a negative small no. and then by another small no. and so on, we get bigger and bigger negative no. $$\dfrac{x}{-small\times small\times.....}=-big\times big\times.....$$
However while doing this, two facts should be noted:
(1) The small no. in the denominator of LHS will never reach $0$.
(2) The big no. in the RHS will never reach $\infty$ or $-\infty$.
Thus by reasoning, if we ever imagine that the small no. in the denominator of LHS is $0$ (which is impossible), we should also imagine that the positive or negative big no. in the RHS is $\infty$ or $-\infty$ respectively (which is also impossible). Is it a reasonable proof for "dividing a no. by zero, we get positive or negative infinity i.e. $\frac{x}{0}=\infty$ or $-\infty$"?
Now by algebraic manipulation,
$$0\times\infty=x$$(where $0$ is positive infinitesimal) $$AND$$ $$0\times-\infty=x$$(where $0$ is negative infinitesimal)
But is algebraic manipulation legal here?
Here, two facts should be noted:
(1) We can never add zero infinite times in positive or negative no. line
(2) We can never get $x=$ non-zero when we keep adding zero over and over.
Thus by reasoning, if we ever imagine that we added zero infinite times (which is impossible), we should also imagine that $x=$ non-zero (which is also impossible). Is it a reasonable proof for "infinity times zero is non-zero i.e. $0\times\infty=x$"?
If it is true, then by algebraic manipulation of $0$ and $\infty$, we are getting logically correct answers.
Now we observe $0\times\infty=x$ could be any no. and hence is undefined. If we want to make $x$ a well defined no., we should make no. lines of $0$ and $\infty$ as an extension of real no. line and give several values for $0$ and $\infty$ like:
$$OR$$
Similarly:
Now if we choose particular values of $0$ and $\infty$, we get a well defined value of $x$. For example:
$$(0\times2)\times(\infty\times3)=(\infty^{-1}\times2)\times(\infty\times3)=2\times3\times\infty^{-1}\times\infty=6$$
Now can we consider this no. line of $0$ as the infinitesimal no. line of non-standard analysis?
In a similiar way, can we also define no. lines of infinitesimal of infinitesimal (second order differentials) and of infinite of infinite and so on?
Am I anywhere incorrect in my reasoning? Are my conclusions logically correct?
Congratulations! You just re-invented Abraham Robinson's framework for analysis with infinitesimals. Note that denoting an infinitesimal by $0$ or more precisely by $o$ has a respectable history starting at least with Isaac Newton. Furthermore, Euler tended to refer to infinitesimals as "exactly zero".
Be that as it may, it turned out to be more productive to use a different notation for them, for example $\epsilon$ and $\delta$ for typical infinitesimals and $H$ and $K$ for typical infinite numbers; see Keisler's wonderful textbook Elementary Calculus for details.
Other than that, the formalism you outlined is perfectly valid, and is much better received by the students than the traditional epsilon-delta approach; see this recent study.