Let me unnecessarily complicate things with these completely idiosyncratic definitions:
I'll start with grief as the suffering of the soul caused by the apprehension of some evil.
Misery is grief when the apprehended evil befalls oneself.
Sympathy is grief when the apprehended evil befalls another person.
Compassion is sympathy that prompts one to help the one who is in misery.
Mercy is the virtue though which compassion is regulated by reason and one acts to help others in misery.
These distinctions allow me to make the following claims:
Compassion is not always a sufficient response. There are circumstances in which a Christian sins if he doesn't act to help someone in misery.
The virtue of mercy can (and probably should) engender compassion in the soul. While it's not, strictly speaking, necessary to feel compassion in order to act with mercy, compassion is a natural human response to the apprehension of another's misery.
Alleviating one's own misery is not in itself an act of mercy. The wicked judge in the parable alleviated the widow's misery not out of mercy, but out of self-love. Relatedly, one can can act to alleviate a shared evil suffered by both oneself and another without acting mercifully toward the other, just as one can suffer the evil without feeling compassion toward the other.