This will sound like nit-picking, but isn't: I'm focusing on the underlying concept, and there is an important concept-level distinction here.
I'm trying to say that there are fundamental concepts related to "number" at play here: the number itself, its
representation, and its
approximation. For many irrational numbers, we can
represent them exactly in many types of computations on a computer, but can only
approximate them in output useful for us humans.
There is some minor merit in the OPs position. In terms of computing there is no way to correctly represent either irrational numbers or infinities.
Not exactly correct.
We can correctly represent many irrational numbers. For example, \$\sqrt{2}\$ as \$2^0.5\$. To convert to a decimal or binary form, we can only show an approximation, but internal calculations can be done exactly, using this and various other representations.
(Background: I have used this exact trick for regular cubic (SC, FCC, and BCC) lattice exact distance histograms. The distances along each coordinate axis are always integers (or halves), so the distance is an expression of form \$d = \sqrt{\sum_{k=1}^N x_k^2}\$ (possibly multiplied by \$2^{-N}\$). Because the distance is nonnegative, we can do all computation in the squared domain, i.e. using \$s = d^2 = \sum_{k=1}^N x_k^2\$, and only use the approximations in the labels when displaying the data. Essentially, my histogram had variable-width bins, but each such bin was exact. Thus, the histogram itself was exact, and only needed to be approximated when output in a form useful to us humans.)
Similarly, arithmetic infinities can easily be represented; even standard IEEE 754 floating-point types do so correctly (in the arithmetic domain).
The expressions and concepts that require irrational numbers can be adequately computed by rational approximations to both real and imaginary numbers.
The expressions and concepts that involve irrational numbers can be adequately
approximated, yes. But, we can also compute many of them in a closed form (including in other arithmetic expressions) that allows us to calculate the approximation to whatever precision we like, only limited by available memory and processing time. \$\pi\$ is a perfect example of this, especially if you want to calculate its representation in binary or hexadecimal instead of decimal, because the
Bailey–Borwein–Plouffe formula exists, making it trivial to calculate the
k'th hexadecimal digit of \$\pi\$ without having to know any of the preceding or succeeding digits.
This does not mean we cannot use entities that
represent the irrational number exactly in computations. Like I explained for square root of two above, there are ways of representing many irrational numbers –– perhaps not all, like some of the physical constants that might turn out to be irrational; consider the
fine structure constant for example –– using alternate, non-binary/non-scalar/non-rational representations in computations; and only need to be approximated for display or final result, for human use.
To ship the information back to their planet, they engrave an extra line between the original two lines on a Pt-Ir alloy rod corresponding to that fraction.

(It is extra funny when one happens to know that you only need 206 bits to express the size of the observable universe in units of Planck length. If the lines were half a meter apart, there would be fewer than 2³² atoms in the lattice between the marks, so you might be able to store the first four ASCII characters that way. Apologies to TimFox for me trying to explain the joke..

)
No, there are innumerable real numbers, an uncountable amount, which means there is no number – and there can be no number – that expresses how many unique real numbers there are.
And you believe it because some men in position of authority told you so.
Nope, I
proved it for myself deduced the logical necessity of it being so for myself, from simple principles.
Really, it's quite an ironic response to the question how to define real numbers.
Not really. I've seen variants of "Have you stopped beating your wife yet?" that were better posed than any of your questions are. If you include an error in your question, you cannot expect correct answers to conform to your erroneous logic/understanding that lead to posing the question.
Like, take a unit square. Remove all points whose either coordinate is rational. Is the total area of the reminder a real number?
Again, a question with an erroneous concept at its core. By definition, the area of a point is zero. So, no matter how countably many points you remove, the area does not change.