Multiplying the inequalities gives a weird result which is obviously wrong

You haven't reached a false conclusion (yet). You started with the assumption that $1<a<2$ and you concluded with $\frac12<a<4$. That's a true consequence (since the open intervals $(1,2)\subset(\frac12,4)$) and a valid argument to reach it. You'd be in a mess if you tried to reverse those steps, though.

It's obviously TRUE.

If $1 < a < 2$ and $\frac 12 < 1$ then $\frac 12 < 1$ and $1 < a$ so $\frac 12 < a$ must be true.

And $2 < 4$ and $a < 2$ so $a < 2 < 4$ so $a < 4$ must be true. So $\frac 12 < a $ true and $a < 4$ is true so $\frac 12 < a < 4$ must be TRUE.

And $\frac 12 < a < 4$ does not contradict $1 < a < 2$.

It's just that $1<a < 2$ is a stronger more precise statement.

Consider this.

$a= 36$ is an even number.

$a=36$ is a perfect square.

$a=36$ is a multiple of $3$.

Fred comes and says "$a$ is a multiple of $3$". Then Jane comes along and says "$a$ is a perfect square". Michael says "$a$ is an even number."

Olav hears all three of these people and says "Wait! Those are three completely different statements! So only one of them can be true! Which one of you is telling the truth and which one is lying."

....

And they are all telling the truth.

$\frac 12 < a < 4$ means that it is possible that $a$ might be less than $1$ but it does not mean that it is. So $\frac 12 < a$ does not contradict $1 < a$.

This is analogous to "$a$ is a multiple of $3$". That means $a$ might be odd but it doesn't mean that it is. "$a$ is a multiple of $3$" does not contradict "$a$ is an even number".

So on.

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Another thing to note:

(Assume only positive values)

$a < b$ and $c < d \implies ac < bd$. But that is one directional. It doesn't go the other way that $ac < bd \not \implies a< b$ and $c < d$

So $1< a < 2\implies (1 < a^2 < 4$ and $\frac 12 < \frac 1a < 1)\implies \frac 12 < a<4$

That is true. And indeed $1< a < 2 \implies \frac 12 < 1 < a < 2 < 4\implies \frac 12 < a < 4$.

But it doesn't go the other way!

$\frac 12 < a < 4 \not \implies 1 < a < 2$

Snd $\frac 12 < a <4 \not \implies (1 < a^2 < 4$ and $\frac 12 < \frac 1a < 1)$

[although $(1 < a^2 < 4$ and $\frac 12 < \frac 1a < 1)$ does actually imply $1 < a < 2$.)

The first two inequalities are, in fact, biconditional ($\iff$), assuming $a > 0$. But when you multiply them, you weaken your final inequality. As you can see, your result $1/2 < a < 4$ is not wrong, but the $\impliedby$ part is. Why would it be? Try to argue and see the flaw in your argument.

"$\iff$ inequalities" have to be manipulated carefully. Losing strength is really easy: $0 < x <1 \implies -0.5 < x < 1.5$, but $x = 1.2$ solves the second one but not the first. You did a lot of implicit steps when multiplying those; if you had $-2<a<-1$, you wouldn't take squares nor reciprocals as easy as you did, and multiplying the inequalities would need aditional steps since they have negative numbers. That's why step-by-step calculations are necessary for strong inequalities.