Learn Spanish (Latin America): Señor, usted no tiene la computadora.

Because it uses usted, the formal word for "you". In Spanish, usted always takes the same verb forms as "he" or "she" (él or ella). If the speaker were talking to a friend, they would use the informal tú and the verb would change to tienes.

Would it be wrong to say "Señor, tú no tienes la computadora"?

Yes, it would sound very awkward. Señor (Sir) sets a formal, respectful tone, so you need to use the formal "you", which is usted. Mixing a formal title with the informal tú sends mixed signals!

Where exactly does the word no go when making a negative sentence?

In Spanish, the word no almost always goes right in front of the verb. Unlike English, which uses helper words like "do not", Spanish just puts no directly before the action. Here, it goes right before tiene.

I've heard another word for computer in Spanish. Why use computadora here?

Computadora is the standard word used in most of Latin America. If you are learning Spanish from Spain, you will usually hear el ordenador instead. Both mean the same thing, but la computadora is the standard Latin American choice.