Learn Finnish: Sain vihdoin painavan vivun siirrettyä. - I finally got the heavy lever moved.

This is the special saada + object + passive past participle structure you are learning. When you manage to get a task completely done, you use the passive past participle (like siirretty, meaning "moved") and put it in the partitive case (siirrettyä). It literally means you "got the lever moved."

If siirrettyä is in the partitive, why is the object painavan vivun in the genitive?

The object follows standard rules based on the main action. Since you completely finished the action of getting the lever moved, it is a total object. Therefore, painava vipu takes the accusative case, which looks exactly like the genitive, ending in -n.

What is the base verb for siirrettyä, and how does it differ from siirtyä?

The base verb is siirtää (to move something else). It is different from siirtyä (to move yourself or for something to move on its own). Because you are acting on an object (the lever), we use the passive participle of siirtää, which is siirretty, and add the partitive -ä for this specific structure.

Could I change the word order to Sain vihdoin siirrettyä painavan vivun?

Yes, you can. Finnish word order is flexible. Putting the participle siirrettyä at the very end emphasizes the completion of the action itself (you got the lever moved), but putting it right after the adverb is also perfectly natural and common.