Proof by contradiction is simple. It is not so much that if a false statement is true, then we arrive at a contradiction. Rather, a better way to think of it is, "if a given statement is true--a statement whose truth or falsity is not yet established--then we arrive at a contradiction; thus the original statement cannot be true."
Take your proof of the irrationality of $\sqrt{3}$ as an example. At the outset, we have not yet established whether or not $\sqrt{3}$ is irrational (or rational). And we don't know until the proof is complete. But you certainly can reason about it by first supposing that if $\sqrt{3}$ were rational, then there exist two positive integers $p, q$ such that $q$ does not divide $p$, for which $p/q = \sqrt{3}$. That follows from the supposition of rationality. Then by continuing the logical chain, you arrive at a contradiction. So if all the consequential steps from the original supposition are valid, but the conclusion is inconsistent, then the original supposition could not be true. Note that in this chain of reasoning, at no point do we actually claim that the original supposition is true, because as we have taken care to mention, we do not know if it is or not until the proof is complete. We are merely saying that if it were the case, we would arrive at a contradiction.
Here is a less trivial example. Some conjectures remain open in the sense that we do not know what the answer is. Consider the Collatz recursion $$x_{n+1} = \begin{cases} 3 x_n + 1, & x_n \text{ is odd} \\ x_n / 2, & x_n \text{ is even}. \end{cases}$$ The conjecture is that for every positive integer $m$, the sequence $\{x_n\}_{n=1}^\infty$ with $x_0 = m$ contains $1$ as an element in the sequence.
If we were to attempt to prove this conjecture is TRUE by contradiction, the supposition we would presumably start with is that there exists a positive integer $m^*$ such that if $x_0 = m^*$, the sequence $\{x_n\}_{n=1}^\infty$ never contains $1$, and then by using some as-yet-unknown logical deduction from this supposition, if we can arrive at a contradiction, we then have shown that the conjecture is true.
But note what I said above: this is an open question! Nobody knows if it is true or false. So you cannot claim that my logic is invalid because I have a priori assumed something that I already know not to be the case!
Now, if I were to desire to disprove the conjecture, it would suffice to find just such an instance of $m^*$ as I have described above. But an extensive computer search has turned up no such counterexample. This leads many to believe the conjecture is true, but no constructive method to substantiate that belief. Mathematics--and number theory in particular--contains many examples of claims that appear true but have their smallest known counterexamples for very large numbers.
So, there is my "informal" explanation of proof by contradiction. The essence is that we don't presume to know the truth or falsity of a given claim or statement at the outset of the proof, so we cannot be accused of assuming that which is false. The falsity is established once the contradiction is shown.
You can interpret modular arithmetic in both of the ways you illustrate, but one of them is a lot more common than the other in mathematics.
The one that's universally understood is the one in which
$$
\frac{19}{2} \equiv 2 \pmod{5}.
$$
The reason is that
$$
3 \times 2 \equiv 1 \pmod{5}
$$
so $3$ is the multiplicative inverse of $2$ and
$$
\frac{19}{2} \equiv 19 \times 3 \equiv 2 \pmod{5}.
$$
In that context you would never write $19/2$ as the decimal $9.5$.
Moveover, in that context expressions like $\pi \pmod{5}$ make no sense at all and some that seem to are impossible. For example, $19/2$ makes no sense modulo $6$ since $2$ does not have a multiplicative inverse modulo $6$.
The other way modular arithmetic is sometimes construed (in computer languages rather than pure mathematics) is as the remainder when you subtract the largest possible multiple of the modulus. I don't want to use $\equiv$ to write that because it bothers my mathematical sensitivity so I will use $\%$ as do some programming languages. Then
$$
19.5 \ \% \ 5 = 4.5
$$
and, as you say,
$$
\pi \ \% \ 2 = \pi - 2 = 1.14159\ldots
$$.
Best Answer
Your student seems to be assuming the following: if a number $n$ can be written as a product in which at least one factor is irrational, then the number $n$ is irrational. Of course, this is not true, and you have found a very instructive counterexample.