Learn Finnish: Myrskyn alettua me katselimme noita rekiä. - After the storm began, we watched those sleds.

This is the past temporal construction. It replaces a subordinate clause like Kun myrsky oli alkanut. Because the storm is a different subject than the "we" in the main clause, "storm" takes the genitive case (myrskyn), and the verb alkaa (to begin) takes the past passive participle form with a partitive ending (alettua).

Why do we use the passive participle alettua if the storm actively began?

That is simply how this specific temporal construction works! When the temporal clause has an explicit subject that is different from the main clause, the action is always expressed using the passive past participle plus the partitive -a/-ä ending, regardless of whether the original action was active or passive.

What is the difference between katsoimme and katselimme here?

Katselimme comes from katsella, which is a continuous verb derived from katsoa (to look). While katsoimme might imply a single, completed glance, katselimme emphasizes an ongoing action over a period of time, which translates perfectly to "we watched".

Why are the sleds (noita rekiä) in the partitive case?