Learn Russian: Анна пишет четыре слова.

After the numbers 2, 3, and 4, Russian requires the noun to be in the Genitive singular case, not the plural! For the neuter noun слово (word), the Genitive singular is слова. It just happens to look exactly like the Nominative plural for this specific word, but grammatically, you are using the Genitive singular rule here.

Why is there a ш in пишет? The dictionary form is писать.

The verb писать (to write) has a common consonant change when conjugated. The с changes to a ш in the present tense, giving us пишет (she writes). This happens in all forms of the present tense, such as я пишу (I write).

Is the phrase четыре слова in the Accusative case because it is the direct object?

Yes! Because Anna is writing them, the four words are the direct object. For inanimate objects (non-living things), the Accusative form of четыре looks exactly like its Nominative dictionary form. The noun слова is locked into the Genitive singular by the number 4, so it doesn't change to a separate Accusative form.

Can Анна пишет mean both "Anna writes" and "Anna is writing"?

Exactly. Russian doesn't have a separate continuous tense like English does. Анна пишет translates to either "Anna writes" or "Anna is writing" depending on what makes the most sense in the English context.