Learn Turkish: Sarhoş adamın aniden herkese küfredesi geldi. - The drunk man suddenly felt an urge to swear at everyone.

In the -(y)AsI gelmek structure, the literal translation is that a person's urge to do something came. Because it is his urge to swear, the drunk man takes the genitive case -ın (adamın), and the action gets the matching 3rd person singular possessive suffix -i (küfredesi).

Why is küfredesi spelled with a d when the base verb is küfretmek?

This is a standard consonant mutation. The helper verb etmek ends in t. When you attach a suffix that starts with a vowel, like the inclination suffix -esi, the t softens to a d (küfredesi).

Why does herkes take the dative case -e (herkese)?

The verb küfretmek (to swear at) always demands the dative case for the target of the swearing. Even though the verb is wrapped up in the -esi geldi pattern, it still dictates the case of its objects exactly like a regular verb would.

The lesson mentions we can use var instead of gelmek. Could we say küfredesi vardı here?