For example if Falkenrath Noble was tapped and another creature dies then would his triggered ability still trigger or only if it was not tapped?
Yes. Whether an permanent is tapped or not has no effect on its abilities. This is the case for abilities of all kinds (static, triggered and activated).
Three things change when a creature is tapped:
- It is considered tapped by anything that cares.
- It cannot be tapped. (Effects instructing to tap the permanent are skipped. Costs consisting of tapping the permanent can't be payed.)
- It cannot attack.
Do note that the first two items can indirectly prevent abilities from triggering ("Whenever an untapped creature dies, ...") or from functioning ("..., tap it. If you do, ...").