Learn Russian: Никто не хочет идти в кино.

Yes, and that is exactly how Russian works! Unlike in English, where a double negative cancels itself out, Russian requires you to use the negative particle не before the verb whenever you use a negative word like никто (nobody), ничего (nothing), or никогда (never). You must have both.

Does никто always take the 'he/she' form of the verb?

Yes. Even though 'nobody' might conceptually refer to a whole group of people, the pronoun никто is grammatically singular and masculine. That is why the verb is хочет (he/she wants) and not хотят (they want).

Why use идти for 'to go' here instead of another verb of motion?

Идти means 'to go on foot' in one specific direction. When talking about going to places in your town like the cinema, a restaurant, or the park for a specific outing, Russians usually use идти, treating it as a one-way trip to a destination.

What case is в кино in, and why didn't the word change?

It is in the accusative case because it shows the direction of your motion (going to the cinema). The word кино does not change because it is a borrowed foreign word. In Russian, many borrowed nouns ending in -о, -и, or -у are 'indeclinable', meaning they never change their endings in any grammatical case.