Learn Arabic (Modern Standard): كان جدي كريم الأخلاق.

It acts as the predicate (or news) of كان, which always forces its predicate into the accusative case. Even though كريم is the first word of an Idafa construct, it still takes the case required by its grammatical role in the sentence.

How can an adjective like كريم be the first word in an Idafa (genitive construct) with الأخلاق?

This is a structure often called a false Idafa or adjective Idafa. It links an adjective (كريم, generous or noble) to a definite noun (الأخلاق, morals) to specify exactly what the adjective applies to. Literally, it means generous of morals.

If جدي (my grandfather) is the subject of كان, why doesn't it show the expected nominative case ending (damma)?

The case marker is hidden by the possessive suffix ـي (my). Whenever a noun attaches to this suffix, the final letter of the noun must take a kasra to blend smoothly with the long 'ee' sound. The nominative damma is grammatically present, but invisible in pronunciation and writing.

The translation says 'character' in the singular, so why does the Arabic use the plural word الأخلاق?

الأخلاق is the plural of خُلُق (trait, manner, or moral). In Arabic, when referring to someone's overall disposition or character, it is standard to use the plural form to represent their entire collection of moral traits.