I have seen arguments such as this $\infty .0 = 0 $ or $\infty .0 = \infty$ and I have seen people disregard this kind of operation by saying that infinity is a concept and not a number so its illogical to perform operations with it. But we use infinity in so much other limits cases in physics and mathematics.
Like is the notion of saying $$\lim_{x \to \infty} f(x) $$ What is the meaning of infinity here? If that x approaches an arbitrarily large number we can always imagine a number greater than that.
What does infinity really signify and why do we in some cases not agree with using infinity and what does it exactly mean by a number approaching infinity ?
For most ordinary math you have that the operations of addition and multiplication are defined for the real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$. Since$\infty$ is not a real number it isn't immediate how you would extend the definitions of addition and multiplication to include $\infty$. You can in a way consider the set $\mathbb{R}\cup\{\infty\}$ and extend the operations. In certain cases we do actually do this. One of them is measure theory we consider the non negative reals with infinity $\mathbb{R}_{\geq 0}\cup \{\infty\}$ or $[0,\infty]$. In this context it makes sense to extend addition such that for all $x$ that $x+\infty=x+\infty=\infty$. You lose some algebraic properties when you do this, for example you can no longer have subtraction being well defined. You also have that the complex numbers with infinity form the so called Riemann Sphere on which for example all polynomial function extend naturally on.
Secondly there are certain "infinite" numbers such as cardinal numbers and ordinal numbers or hyperreal numbers for which the operations of addition and multiplication are well defined. These contexts are a bit different since you have many infinite numbers. So in no way is it illogical to have operations involving infinity. I would be better to say that their definition heavily depends on the context in which you are working in and in most cases are not defined.\
For the second part of your question is less algebraic and more topological. An "infinite" element may be added to a topological space to formalize the definition of a sequence "escaping" the space or becoming arbitrarily large. For example we have that the set $[0,+\infty]$ has a natural topology which makes it homeomorphic to $[0,1]$. A sequence of positive reals $(r_n)_{n\in\mathbb{N}}$ becomes definitely arbitrarily large or rather: for all $M$ there exists $N$ such that for all $n> N$ we have $r_n>M$ if an only if the sequence $(r_n)_{n\in\mathbb{N}}$ converges to $+\infty$ in the space $[0,+\infty]$. Similarly we have that a sequence on the Riemann sphere converges to $\infty$ if the modulus of the sequence becomes definitely arbitrarily large.
To summarize, $\infty$ is usually used for convenience and its use varies from context to context. Many times it is used in hand wavy arguments or to help give some intuition as to what is happening. One last example is that if we have a sequence $(r_n)_{n\in \mathbb{N}}$ such that $\lim_{n\rightarrow \infty} r_n=+\infty$ and another sequence such that $\lim_{n\rightarrow \infty} t_n=t$ where $t$ is a real number then we have $\lim_{n\rightarrow \infty} r_n+t_n=\lim_{n\rightarrow \infty} r_n+\lim_{n\rightarrow \infty} r_n=+\infty+t=\infty$. This gives a better picture of what is occuring then simply using limit definitions and is why sometimes a more hand wavy argument gives us better insight as to what is happening.