= Solution
Modulo $3$, the affine points are
$$
(0,1),\quad(1,1),\quad(2,1).
$$
Each equals its own inverse because $-1-1=1$ in $\mathbb F_3$. Thus
$$
\widetilde E(\mathbb F_3)\cong(\mathbb Z/2\mathbb Z)^2,
$$
which is noncyclic. The reductions of $P$ and $Q$ are the distinct nonzero points $(0,1)$ and $(1,1)$, so they form a basis.
If $mP+nQ=O$, reduction modulo $3$ shows that $m$ and $n$ are even. Write $m=2m_1$ and $n=2n_1$. Then $m_1P+n_1Q$ is a rational point of order dividing two, and part (iii) makes it zero. Repeating the argument shows that $m$ and $n$ are divisible by every power of two, so $m=n=0$. Therefore
$$
\boxed{P\text{ and }Q\text{ are independent points of infinite order}.}
$$
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