Weapons don't break down but the enchants lose charge and need a soul gen to recharge
Armor breaks down and can be repaired by any merchant for gold or by repair kits sold by merchants or given with daily writ rewards
Do weapons lose damage gradually while breaking down? And does armor lose defence meanwhile?
Armor used to but it's no longer the case. Even if your armor is at 1/3000 it still will be providing full 3000 defence, armor passives and set boni. Once it goes down to 0 it will stop providing the defence and I think passives but will still provide set boni(or vice versa, I always forget if it's set boni but no passives or the other way round>.<).
Weapons don't break down, like the poster above said, they might only lose charge.