Learn Turkish: Çocuk oyuncağı kaptı ve kaçtı. - The child grabbed the toy and ran away.

You would drop the definite past tense (-tı) and use the aorist (-ar / -er). As a sudden, first-person decision, it becomes: Oyuncağı kaparım ve kaçarım! If you were warning someone about what the child might suddenly decide to do, you'd use the third person: Çocuk oyuncağı kapar ve kaçar!

Native speakers often avoid using the word ve (and) when combining two sequential actions. How could we rewrite this sentence to sound more natural?

You can use the -ıp / -ip connecting suffix on the first verb instead of conjugating it. The sentence becomes Çocuk oyuncağı kapıp kaçtı. This is a very common B2-level structure to show that one action happened immediately after another, translating roughly to 'The child, having grabbed the toy, ran away.'

Could I just use aldı (took) instead of kaptı (grabbed) here?

You could, and the grammar would be perfectly fine (Çocuk oyuncağı aldı ve kaçtı), but you would lose the descriptive nuance. Kapmak specifically means to snatch, seize, or grab something forcefully or quickly, whereas almak is a neutral 'to take.' Using kaptı fits the mischievous nature of the action much better.

I know we add the accusative case for specific objects, but as a quick refresher, why did the k in oyuncak disappear?