How is metal fatigue detected in an aircraft?
How do the engineers, maintenance team, etc know if an aircraft has metal fatigue?


First of all, is not an "aircraft" that has fatigue, but a component.

To the naked eye, there is no way to detect metal fatigue until it starts being too late: you can only see cracks that are already forming and/or propagating.

There are tools available to allow for early detection, my bachelor's thesis was about one of these: the photoelasticity of certain materials is used. A light coat of one such material is applied on the new component and the photoelastic properties and exams done during the lifetime of the component tell the maintenance people what is the current status of the inner layers of the piece.

Another method uses electric resistance, as metal fatigue, by cracking the metal, increases the resistance.

Depending on the scenario, X-rays could also be used.


  What do they do to "minimize" metal fatigue or repair it?


You cannot minimize fatigue, once it starts setting in, it is an irreversible process. You can delay the likely catastrophic failure by applying strengthening elements around the affected one, but the safest course of action is to replace the component and recycle the old one.