Learn Finnish: Älä vedä koiraa! - Do not pull the dog!

The primary reason is that this is a negative sentence, triggered by älä (don't), and negative sentences require a partitive object. Additionally, the verb vetää (to pull) typically takes the partitive anyway because pulling is usually an ongoing, continuous action.

Where does the form vedä come from, and why is there a 'd' instead of a 't'?

It comes from the verb vetää (to pull). For negative commands directed at one person, we use älä plus the verb's weak stem. You can find this stem by taking the first-person form (vedän) and dropping the -n. The t weakens to d because of consonant gradation.

This lesson is about phase verbs. How would I change this sentence to mean 'Stop pulling the dog!' using lakata?

You would use the imperative of lakata (lakkaa) followed by the 3rd infinitive elative form of vetää, which is vetämästä. The full sentence becomes Lakkaa vetämästä koiraa!