They can (and have been) called in to quell violent street unrest, looting etc, by show of armed force, but against civil crimes like theft and murder they aren't equipped to resolve anything. It is not at all how crime prevention works (to say nothing of Congress taking away most of the DC budget).
I grew up on the Texas Gulf Coast and the most common reason for the governor to deploy Guardsmen was before and after hurricanes, to help communities build up supplies, put up sandbags, and help with the cleanup after, guard damaged stores where the glass had blown out etc. Helping hands basically.
So a bit like our THW, I guess. But he's trying to make them into his army for use internally, which is worrying.
In normal practice, yes, though strictly speaking they are an armed military reserve force, and I don't think about the THW in that way. They were intended as a supplement to the active armed forces in wartime, and do in fact deploy in combat situations. But mostly it is civil assistance projects.
They are often called up during wartime and National Guard units were deployed to Iraq and Afghanistan, ultimately providing 1/2 the overall troop numbers. I also don't think they have the specialized rescue and disaster relief training the THW does. www.pbs.org/newshour/sho...
Yes, primarily meant as an overseas combat reserve force, but often used domestically to support civilian emergency services. Not at all what Trump is trying to present as a sort of domestic national police force under White House control. That's explicitly what the judge in the LA case struck down.