Learn Russian: Пускай король слышит этот тост!

Both words mean "let" and are used to create a command for a third person (him, her, them). Пускай is just a slightly more casual, conversational version of пусть. You can use them completely interchangeably here.

Why do we use the regular present tense verb слышит instead of a special command form?

Russian doesn't have a unique third-person imperative verb ending. Instead, you just use пусть or пускай followed by the regular present or future tense. Since we want the king to hear the toast right now as it is being spoken, we use the present tense слышит.

Why does the infinitive слышать end in "-ать", but this form is spelled слышит with an "и"?

The verb слышать is a famous exception. Even though it ends in "-ать", it conjugates according to the second conjugation pattern (which usually has "-ить" endings). That is why it uses an "и" in forms like он слышит instead of an "е".

Why doesn't the direct object этот тост change its ending here?

It is actually in the accusative case because it is receiving the action of the verb. However, because тост is a masculine inanimate noun (a non-living thing), its accusative form looks exactly the same as its dictionary form. The word этот follows the exact same rule.