Learn Finnish: Sain vihdoin tämän ruuvin kiinnitetyksi seinään. - I finally got this screw attached to the wall.

This is the standard written form of the 'getting something done' construction being taught in this lesson. It uses the verb saada + object + the passive past participle in the translative case. The translative case expresses the end result of an action: you achieved the state of the screw being attached. Note that in everyday spoken Finnish, you will often hear the partitive form (kiinnitettyä) used instead of the translative.

Why is the object tämän ruuvin in the genitive-like accusative case?

Because the action was fully completed. The construction saada + translative participle intrinsically implies successfully finishing a task. Since you achieved the goal of attaching the entire screw, it takes a total object, which here ends in -n. If the action were somehow incomplete or negative, you would use the partitive case.

In English we attach a screw 'to' a wall. Why does Finnish use the illative (into) case seinään?

When attaching, connecting, or fixing something firmly to a surface, Finnish conceptualizes the movement as going into the destination, hence the illative case. This makes perfect physical sense for a screw penetrating the surface, but remember that verbs of attaching (like kiinnittää) generally require this 'into' or 'onto' directional case for the location.

What is the base verb that kiinnitetyksi comes from?