Learn Russian: Мальчик хотел бы взять этот сок.

The combination of a past tense verb (хотел) and the particle бы creates the conditional mood. Together, they translate to "would like" instead of just "wanted" or "wants". In Russian, this is the standard way to express a polite wish or request in the present moment.

Does the particle бы always have to come right after the verb?

Not always, but it is the most common and natural spot. The particle бы is flexible and often attaches to the word you want to emphasize. For example, you could say Мальчик бы хотел..., but you can never put бы as the very first word in a sentence.

Why use the perfective verb взять here instead of the imperfective брать?

The boy wants to take the juice as a single, completed action. When making a polite request for a specific, one-time result ("to take this juice"), Russian typically uses perfective verbs like взять.

I know the verb взять requires the Accusative case for the object being taken. Why does этот сок look like the dictionary form?

It actually is in the Accusative case! The noun сок (juice) is masculine and inanimate (not alive). For inanimate masculine nouns, the Accusative case form is exactly the same as the Nominative (dictionary) form. The pointing word этот (this) follows the exact same rule.