Learn Russian: Как твоя сестра пишет этот текст? Хорошо? - How is your sister writing this text? Well?

Хорошо (well) is an adverb, which means it describes an action. Here, it asks about how the sister writes (пишет). Words like хороший or хорошая are adjectives (meaning "good"), which describe nouns, like a "good sister" (хорошая сестра).

Why is there a "ш" in пишет? Isn't the verb писать?

Yes, the infinitive is писать (to write). However, this verb has a common stem change when conjugated in the present tense: the с changes to ш. Because "your sister" is a "she" (3rd person singular), it becomes пишет instead of "писает".

The text is receiving the action. Why didn't этот текст change its ending?

You are right that the object of an action goes into the accusative case. However, for masculine nouns that are non-living (inanimate) like текст, the accusative form looks exactly the same as the dictionary (nominative) form. The word этот (this) also stays the same.

The English translation says "is writing", but isn't пишет just "writes"?

Russian doesn't have a separate "present continuous" tense like English does. The single word пишет covers both "she writes" and "she is writing". You can tell from the context of this sentence that the speaker is asking about an action happening right now.