A proportionality puzzle: If half of $5$ is $3$, then what's one-third of $10$?

From a false assumption you can derive anything. Answer what you want: it will be correct.

For example: the answer is $\pi^2$, and I'll prove it. Suppose not. Then, by hypothesis, $5/2=3$, so $5=6$ and, substracting $5$ to each side of equation, $0=1$, a contradiction. So the answer is $\pi^2$.

I think this is more a question of language than of mathematics. (Indicated also by the fact that a "foreign country" is mentioned.)

A possible understanding of "half" in this case would be that "half" is an operation that assigns integers to integers by splitting them in to parts as evenly as possible and then taking the largest part. In other words, by "half" of $x$ could mean the smallest integer that is not less then half (with its usual meaning) of $x$, which we usually denote $\lceil\frac{x}{2}\rceil$.

Based on this same understanding, a "third" of $10$ would mean $\lceil\frac{10}{3}\rceil$, which is $4$.

But the result you will get in the end will ultimately depend on the way of thinking in that country.

As $5/2=3;$
it implies, $5/3=2$;
and $2*5/3=2*2=4;$
hence $10/3=4;$

I imagine this to be a problem caused by this foreign country not having the concept of the number zero.

If you think about it as a number line, without a 0:

If you were to divide this line into two equal parts, you would draw a line through the tick that corresponds to the number 3. Therefore, you could say that "half of 5 is 3"

The same goes for a number line that includes 1 through 10. If you wanted to divide that line into 3 equal parts, you would draw lines through the ticks that correspond to the number 4 and the number 7. Therefore, you could say that "One third of 10 is 4" and "two thirds of 10 is 7" which seems internally consistent because you could also claim that "one half of 7 is 4."

Of course, this makes no sense and only shows up because this country apparently doesn't consider any numbers less than 1.

I think your teacher wanted the following solution in which we only use the given relation that $\frac{5}{2}=3$: $$ \frac{10}{3} = \frac{4}{3}\frac{5}{2} = 3\frac{4}{3} = 4. $$