Learn Italian: Sono lontano e ho nostalgia di casa. - I am far away and I am homesick.

In Italian, many physical and emotional states are expressed with the verb avere (to have) followed by a noun, rather than essere (to be) plus an adjective. Just like you say ho fame (I have hunger = I am hungry), you say ho nostalgia (I have nostalgia = I am homesick).

If a woman is speaking, would she say Sono lontana instead of Sono lontano?

Yes! In this sentence, lontano is acting as an adjective describing the subject. Therefore, it must agree in gender and number. A woman would say Sono lontana, and a mixed or all-male group would say Siamo lontani.

Why is it just di casa and not della casa?

When referring to the concept of 'home' rather than a specific physical building ('the house'), Italian drops the definite article. You see this in common phrases like andare a casa (to go home) or essere a casa (to be at home). So, 'homesickness' is simply nostalgia di casa.

Could I use the verb mancare (to miss) here instead?

You could, but the grammar would change completely. To say 'I miss home' using mancare, you would say Mi manca casa (literally: 'Home is missing to me'). However, avere nostalgia di casa is the standard, most natural way to express the specific state of being 'homesick'.