Learn Russian: Я приду на лечение через неделю. - I will arrive for the treatment in a week.

The preposition через combined with a time word in the accusative case is how Russian expresses that something will happen "in" or "after" a certain amount of time. The dictionary word is неделя (week), and its accusative form is неделю.

Why do we use на лечение instead of для лечения if the English says "for the treatment"?

While для does mean "for," Russian uses the preposition на + the accusative case when you are going to an event, activity, or process. Since a medical treatment is a scheduled activity you attend, we use на.

Is лечение in the accusative case here? It looks like the dictionary form.

Yes, it is in the accusative case because it follows the preposition на indicating the destination or purpose of your movement. Neuter nouns that end in -е do not change their spelling in the accusative case, so it looks exactly like the dictionary form.

Why does приду mean "I will arrive" rather than "I am arriving"?

Приду is a form of the perfective verb прийти (to arrive on foot). Perfective verbs describe completed actions, so when you conjugate them, they automatically refer to the future rather than the present tense.