Learn Russian: Грязная вода проходит через фильтр.

The preposition через requires the accusative case. For masculine inanimate nouns like фильтр, the accusative form is exactly the same as the nominative (dictionary) form. If the water were passing through a feminine noun like труба (pipe), you would see a change: через трубу.

Why do we need the про- prefix on проходит if we already have через meaning "through"?

In Russian, it is very common to pair a prefixed verb of motion with a preposition that echoes its exact meaning. The про- prefix means "through" or "past." While you could technically use a basic verb, combining проходить and через sounds much more natural and specific for the action of passing through something.

Could I use сквозь instead of через here, since the water is penetrating a solid material?

Yes, you could say сквозь фильтр, and it would be grammatically correct. Через is the standard, neutral word for "through." Сквозь specifically emphasizes passing through a dense medium, often implying some resistance or effort. For a filter doing its regular job, the neutral через is perfectly natural.

Why is the imperfective verb проходит used here instead of the perfective пройдёт?