Learn French: Vous jetez le grand carton à la poubelle. - You throw the big cardboard box in the trash can.

It does double the "t", but only for certain pronouns! Verbs like jeter (to throw) double the consonant for je, tu, il/elle/on, and ils/elles (like je jette). For nous and vous, they keep the single consonant from the infinitive: vous jetez.

Why does grand come before carton instead of after?

While most French adjectives come after the noun, grand (big) is an exception. Short, common adjectives describing Beauty, Age, Goodness, or Size (often remembered by the acronym "BAGS") go before the noun.

The English translation says "in the trash can", but the French uses à la (to the). Why not dans la?

You can absolutely say dans la poubelle, but jeter à la poubelle is a very common fixed expression. When throwing something away, French speakers often phrase it as throwing it to the trash.

Can carton mean both "cardboard" and "cardboard box"?

Yes! Un carton is the everyday French word for a cardboard box (like a moving box or shipping box), but the word also refers to the material itself, "cardboard."