Aidan T. answered 04/03/20
Geology PhD student
Hi! The typical answer to this is they can be as deep as ~15 km. This is the point at which pressures and temperatures become so great the quartz dominated crust starts to flow rather than break, so below this depth you start to get ductile rather than brittle deformation (folding opposed to faulting). As you rightly said, things like mineralized fluids do fill up the gaps, hence why you see calcite and quartz in some faults. This is the basic answer however...
In places like Thingvellir National Park in Iceland though, you're looking at a very unique case of geology. As with most things in life, there's no one fits all rule. You're looking at thinner crust, because it's oceanic not continental crust, that is rifting apart. Iceland formed due to a hotspot hitting the mid-ocean ridge and so the tectonics are unusual.
Faults can form 'anywhere' within the crust and do not need to intersect the surface, these are called blind faults. Most of these faults are known either through seismic data or earthquake data, that can pinpoint the hypocenter or focus.