Learn Finnish: Kiipeilijät panivat otsalampun päähän.

This is the distributive singular. In Finnish, if each member of a plural subject (the climbers) acts upon something they each have exactly one of (one headlamp, one head), the item and body part are usually put in the singular. English translates this as plural ('headlamps on their heads'), but Finnish prefers the logical singular per person.

Since we are treating the headlamps as a singular object, why does otsalampun end in -n?

Because it is the total object of an affirmative sentence. When we use the distributive singular for the object, it follows the standard rules for singular objects. Since the action of putting on the gear is complete, it takes the singular accusative case, which ends in -n (looking identical to the genitive).

The English translation says 'put on', but what is päähän doing here literally?

Päähän is the illative case ('into' case) of pää (head). In Finnish, when you put clothing or gear on your body, you specify the body part using the illative. Instead of a generic 'on', you literally put the headlamp 'into the head'.

I usually see laittaa used for 'to put'. Is panivat (from panna) old-fashioned?