What is the name of the long staff with an iron tip used in mountaineering

Alpenstock

Trekking poles? For hiking, balance, fording water, and l suppose negotiating your footsteps in the mountains.

Fancy walking stick with a cleat on the bottom for more traction.

Some say they tear up the trails and do not use them.

pike

This would either be a treking pole (modern ones have a carbide tip that lasts longer than metal) or an ice axe use for glacier travel.

Good Luck