Learn Italian: Il maiale che mangia la mela è molto grasso. - The pig that is eating the apple is very fat.

In this sentence, che acts as the subject of the verb mangia. You can tell because it represents il maiale (the pig), which is the one actively doing the eating. If che were the direct object, the verb would need a different subject (for example, il maiale che mangio means the pig that I eat).

Could I use cui instead of che here?

No, you must use che. The relative pronoun cui is only used after a preposition (like di, a, or con). Because che connects directly to the verb as its subject, it is the only correct choice here.

Why is mangia translated as is eating? Shouldn't that be sta mangiando?

The standard Italian present tense (mangia) translates to both eats and is eating in English. While you could use the present continuous (sta mangiando) to emphasize that the pig is eating at this exact second, the simple present mangia is perfectly natural and very common for ongoing actions in Italian.

Would grasso change if the animal were feminine?