You end your discussion with the situation where the garden's soil is wet and it rained. Therefore, $P$ is true and $\neg Q$ is also true. This contradicts your original assumption that $P\Rightarrow Q$ is true.
The only way for $P\Rightarrow Q$ to be true is that whenever it rains, it is impossible for the soil to be wet (even, perhaps, for other reasons). In other words, for $P\Rightarrow Q$ to be true, then when it rains, the rain prevents all possible reasons for the soil to be wet (it's not just that the soil is not wet because of the rain).
Try not to confuse causality with implication. $P\Rightarrow Q$ means that whenever $P$ is true, then $Q$ must also be true; this does not mean that $P$ causes $Q$. Think about this case (you have a class with two exams and no other graded work): $A$ is "I failed the class" and $B$ is "I failed an exam." $A\Rightarrow B$ is true because if you failed the class, then you had to have failed an exam (note that if you fail an exam, you might not have failed the entire class, so $B\Rightarrow A$ is false). In this case, the observation of $A$ allows you to conclude that $B$ happened, but $A$ does not cause $B$. In fact, $B$ partially causes $A$, but $B$ does not imply $A$.
Therefore, neither $P\Rightarrow Q$ nor $\neg Q\Rightarrow \neg P$ are true in this case.
In order to fix your original statements, let $P$ be the event that it rains and $\neg Q$ is the event that the garden is wet because of rain. (Then $Q$ is the garden is not wet or the garden is wet for some reason other than rain.) Then $P\Rightarrow Q$ and $\neg Q\Rightarrow \neg P$ are true because in your final consideration, $\neg Q$ is false, the garden is wet, but not because of rain.