I kind of assume that dadoes, grooves, rabbets / rebates should be no deeper than half the thickness of the material. So, in 3/4" stock, dadoes should be no deeper than 3/8". Anything deeper would compromise the integrity of the wood. Is that correct?
Broadly speaking yes.
In just about any wood, with any joint, if you end up removing more wood than remains you're obviously undermining the strength of the material. So with a dado (UK: housing joint) going deeper than half way should be avoided.
And, going the other way, is there a minimum depth of a dado before it's either not adding any strength, or not assisting with alignment? Maybe 1/8" or 1/16"?
This is less clear cut I feel because so much depends on the inherent strength of the material. While a 1/8" (3mm) dado wouldn't be particularly strong in any wood in stronger hardwoods it could be more than sufficient, while in something like pine the same joint depth would probably represent a noticeable weak point.
The expected load and the outright thickness of the material obviously tie in here as well. In a small campaign case in pine 1/16" dados may be appropriate and provide suitable strength, but obviously not for a pine shelf in a sizeable bookcase or cupboard where you want to give the ends of the boards much more support.
I suppose one way of looking at it is if there's no good reason to do a shallow dado then don't. Many joints in woodwork are actually stronger than strictly necessary, but erring on the side of greater strength is rarely a bad thing. And as it may be no more or less effort to create a weaker version then why do it anyway?
In some of the youtube videos I've seen, it seems that 1/4" dadoes are fairly common in 3/4" material. Is that because it gives the optimum strength, or is just easier / faster to cut, or something else?
I suspect some of this will come down to habit or ease of setup rather than being carefully considered and done with best strength in mind.
But it is actually quite a common rule of thumb to have a joint be 1/3 the thickness of the material, although that's by no means set in stone and variation either way is possible and acceptable.