The above sentence is nominal sentence, formed by a nominal subject and a nominal predicate; Meaning: the speaker is telling the receiver that his father is generous. The adjective (كريم) [generous] here is a "predicative" adjective telling the status of "my father," and hence "my father IS generous".
Now, if I want to describe "my father" as being generous using an attributive adjective, then I would say (أبي الكريم) [abí-l-karím] -using my own notation here- [my generous father]. Notice how (كريم) becomes (الكريم), by adding a definition article to it (الـ). This is because when adjectives becomes attributive (i.e. attached to the noun and describes the noun), the adjective must follow the noun in gender, number and definition. Adding (-í) to the word (i.e. "my") is a form of definition to the word in Arabic, and hence the word (كريم) has to be defined as well, and hence we add (الـ) to mimic the noun which it describes (أبي).
Yep. The predicative in nominal sentences in Arabic are typically and mostly undefined (talking about simple structure here).
The "a" of AL, disappears in pronunciation of أبي الكريم but not in writing of course because it is Hamzat-Wassl (connecting Hamza), so it acts somewhat like a schwa. In fact here in this specific sentence, the word أبي (my father) might have a shorter vowel (-í) to its end when speaking it and connecting it to the word after. Just an approximation of how it would sound: abilkarím. Some might say it as: abiyalkarím.
UW. As a thumb rule (nothing official here): The tendency in Arabic is to base the spelling on sounding the words singularly, and let the tongue do the rest when speaking as things come naturally in sentences. For this, spelling does not change much in Arabic except in limited cases (and mostly for conventional orthography rather than spelling).