Learn Russian: Он просит помощи.

The verb просить (to ask for) often requires the genitive case when the thing being asked for is abstract or uncountable. Since help is an abstract concept, it takes the genitive form помощи.

Why does the genitive form of помощь end in -и?

Помощь is a feminine noun that ends in a soft sign (-ь). Nouns in this specific group (the 3rd declension) drop the soft sign and take the ending -и in the genitive case.

This lesson is about commands. How would I tell someone "Ask for help!" versus "Don't ask for help!"?

To give a positive command, you usually use the perfective aspect: Попроси помощи! (Ask for help!). For negative prohibitions, you use the imperfective aspect: Не проси помощи! (Don't ask for help!). Our example sentence uses the imperfective просит simply because it is an ongoing present-tense statement, not a command.

Could I use the verb спрашивает instead of просит here?

No, because Russian uses two entirely different verbs for "to ask". You use спрашивать when you are asking a question to get information. You use просить when you are requesting a thing, an action, or a favor, such as asking for help.