Learn Italian: I miei genitori adottivi hanno invitato la mia sorellastra. - My adoptive parents invited my stepsister.

The general rule is to drop the definite article with singular family members, so you would normally say mia sorella. However, sorellastra uses an altering suffix (-astra). Altered family nouns always keep the definite article, which is why it must be la mia sorellastra.

Does genitori adottivi take the article i because it is modified by an adjective, or because it is plural?

Both rules apply here! Plural family members always take the definite article, so it must be i miei genitori. Additionally, any family member modified by an adjective, such as adottivi, also requires the article, even if it were singular.

Does the suffix -astra always mean step- when talking about family?

Yes, for family members, the suffix -astro or -astra denotes a step-relative or half-sibling, like fratellastro. Outside of family words, this suffix usually gives a pejorative or weakened meaning to a word, such as giallastro for yellowish.

Why doesn't the past participle invitato agree with the feminine sorellastra and become invitata?