= Solution
For a reduced rational number $x=a/b$ with $b>0$, the <height of a rational number> is
$$
H(x)=\max\{|a|,b\}.
$$
Condition (i) holds because only finitely many coprime integer pairs have bounded maximum.
Condition (iii) also holds. The standard height inequality
$$
H(x\pm y)\leq2H(x)H(y)
$$
gives
$$
h(x+y)+h(x-y)
\leq2h(x)+2h(y)+2\log2.
$$
Condition (ii) fails: for positive integers $m$,
$$
h(2m)-4h(m)=\log2-3\log m,
$$
whose absolute value is unbounded. Thus precisely conditions $\boxed{\text{(i) and (iii)}}$ hold. This is consistent with $\mathbb Q/n\mathbb Q=0$ although the additive group $\mathbb Q$ is not finitely generated.
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