Psalm 58 – Skeptic's Annotated Bible answered

Recent research has shown that:

Whether lying about raiding the biscuit tin or denying they broke a toy, all children try to mislead their parents at some time. Yet it now appears that babies learn to deceive from a far younger age than anyone previously suspected.
Behavioral experts have found that infants begin to lie from as young as six months. Simple fibs help to train them for more complex deceptions in later life.
Until now, psychologists had thought the developing brains were not capable of the difficult art of lying until four years old.

But the psalmist already knew that long before.

The psalmist does not pray he can do those things. He asks for God's judgment on liars. Their mouth should be stopped. Or does the author of the SAB rather have that the lying can continue? We do not pray that our enemies can continue in their evil ways. We also don't ask that we can stop them. But we ask God to either convert them or take them away.

It's sad if one cannot appreciate poetry...

The righteous doesn't rejoice here simply because his enemy is in misery, but because they see the vengeance of God and their delivery out of their enemies hand. Calvin comments:

It might appear at first sight that the feeling here attributed to the righteous is far from being consistent with the mercy which ought to characterize them; but we must remember, as I have often observed elsewhere, that the affection which David means to impute to them is one of a pure and well-regulated kind; and in this case there is nothing absurd in supposing that believers, under the influence and guidance of the Holy Ghost, should rejoice in witnessing the execution of divine judgments. That cruel satisfaction which too many feel when they see their enemies destroyed, is the result of the unholy passions of hatred, anger, or impatience, inducing an inordinate desire of revenge.

See also Prov. 24:17.