$\frac{1}{\infty}$ - is this equal $0$?

The notation $\displaystyle\frac{1}{\infty}=0$ is used as a shorthand for

"As $x$ approaches infinity, the denominator blows up without bound, and hence since the numerator is constant, the value of the function approaches zero (i.e. gets arbitrarily close to zero), and hence its limit is zero."

The notation $\dfrac 1\infty$ does NOT literally mean "divide $1$ by $\infty$".

So literally, it is nonsense; taken as shorthand for the above, you'll see that notation used pretty commonly when folks evaluate limits. It's what we call "an abuse of notation."

Let's say that you want to divide $1$ pie among an $\infty$ of your friends. Each friend will get $\frac{1}{\infty}$ of the pie or, as you say $0$. After you give out the pie, you rethink matters and say to your friends to give the pie back to you. Each friend will give you his portion of the pie that is a $0$. So you end up with $0$ pies. Where has the pie gone?

A rather interesting argument against $0\cdot\infty =1$ occurred to me. Suppose $0\cdot\infty =1$ holds. Then $1 = \infty \cdot 0 = \infty \cdot (0 \cdot 0) = (\infty \cdot 0) \cdot 0 = 1 \cdot 0 = 0$. Strange, isn't it?

Yes and no. Think about the implications. Is $0\cdot\infty=1$ ? Because normally, if $\displaystyle\frac ab=c$ , then $a=bc$ But this isn't really the case here, is it ? Because, since all limits of the form $\displaystyle{\lim_{n\to\infty}\frac kn}$ are $0$, for all finite numbers k, then the product $0\cdot\infty$ becomes meaningless. Sometimes it can even be infinity itself, since $\displaystyle{\lim_{n\to\infty}\frac n{n^2}=0}$, for instance. So it is best to avoid such expressions, especially if you are a beginner. Otherwise you'll soon be asking questions like why $\displaystyle\lim_{n\to\infty}\left(1+\frac1n\right)^n=e$ instead of $1$, since, by all appearances, $\frac1\infty=0$, and $\displaystyle\lim_{n\to\infty}1^n=1$.