Learn Russian: Открой окно, пожалуйста!

In Russian, affirmative commands (telling someone to do something) usually use the perfective aspect when you want a single, completed action with a clear result. You want the window to be opened and stay open, so you use the perfective открой.

How would I say "Don't open the window!" using the rules from this lesson?

You would say Не открывай окно! This lesson teaches that negative commands (telling someone not to do something) almost always use the imperfective aspect. So instead of just adding не to открой, you must switch to the imperfective verb открывать and use its command form, открывай.

How is the command form открой built from the dictionary form открыть (to open)?

To make an imperative, we look at the "they" form of the verb. For открыть, the "they" form is откроют. We drop the -ют ending, leaving the vowel о at the end of the stem. Because the stem ends in a vowel, we add -й for the informal "you" (ты) command, giving us открой.

Окно is the direct object here. Why doesn't its ending change to show the accusative case?

Direct objects do take the accusative case, but окно (window) is a neuter noun. For all neuter nouns, as well as masculine inanimate nouns (objects, not people or animals), the accusative form is exactly the same as the nominative (dictionary) form.