How deep are the cracks in the earths crust at fault zones?
Cracks (actually the right word is fault or in some situations rift) can only occur in brittle material and only the Earths upper crust (top 5-25 km) is brittle. The lower crust and mantle are ductile, i.e., they have a relatively lower viscosity which means that they can indeed crack on short time scales but will heal quickly due to high temperatures and pressure.

Edit: In some ways you can think the lower crust and mantle to be like pitch (tar), which is solid at room temperature but flows over longer periods. E.g., see this link which also happens to be the worlds longest running experiment.