Broad-side swords, crossbows, battle axes and catapults are the common ones. The Trebuchet is a weapon that flung rocks or dead cows hoping to spread disease. catapults were out-classed by the...
Yes. The nobles/knights were required to have practise each week and they worked all the time on their weapons and use of them. Tournaments were for that purpose as well as having fun.
Medieval weapons were stored in armories. An armory could be a separate building, but could also be just a room in a castle, or at a city wall or other fortification.