Learn Finnish: Tuskin olin avannut oven, kun avain lipsahti kädestäni. - I had barely opened the door when the key slipped from my hand.

This is the classic pattern for a tuskin ... kun (hardly ... when) narrative sequence. The pluperfect (olin avannut) sets the stage with an action that had just finished, while the regular past tense (lipsahti) introduces the sudden, immediate event that happened right after.

Does tuskin have to be at the very beginning of the sentence here?

Yes. In this specific dramatic structure, tuskin is placed at the absolute start of the sentence for emphasis. If you were just saying 'I had barely opened the door' without the 'when' clause, you might put it later, like Olin tuskin avannut oven.

Why is the object oven in the accusative form ending in -n instead of the partitive?

Opening the door is treated as a complete action here. The pluperfect tense shows that the door was successfully and completely opened right before the key dropped. Because the action was seen through to completion, the total object rule applies and we use the -n ending.

Why use the elative case (-stä) in kädestäni instead of the ablative (-ltä) for 'from my hand'?

Finnish treats a hand's grip as an enclosed space. When something slips from your grasp, it comes 'out of' your hand, so you use the internal movement case (-stä). If you used the external case (kädeltäni), it would sound like the key merely slipped off the flat surface of your hand.