The correct answer is $\boxed{\text{A. }19, 66}$.
The sequence is $7, 9, 66, 12, 14, 66, 17, \dots$. Each number is followed by $66$, except for $9$, which is followed by $12$. This suggests that the sequence is made up of two alternating subsequences: $7, 12, 17, \dots$ and $9, 66, 66, \dots$. The first subsequence increases by $5$ each time, while the second subsequence is constant. Therefore, the next number in the sequence is $19$, followed by $66$.
Option B is incorrect because $66$ is followed by $19$, not $19$ by $66$. Option C is incorrect because $22$ is not a multiple of $5$. Option D is incorrect because $20$ is not in the sequence. Option E is incorrect because $66$ is followed by $19$, not $20$.