Anyone know if it doubles the HP and stamina these give, or just makes it last longer? Or both, maybe?

I might not bother using it on my dremora lords if it's only 25 health, but if it's 50 that for some reason just seems much more significant to me.

The dual-casting perk for each tree will tell you what the benefit for dual-casting in that particualr school , I believe conjuration dual casting lengthens spell duration.

It tells you but it's incomplete information, since it can do different things for illusion(the school for rally/courage isn't conjuration). It says it overcharges it into a more powerful version for illusion, but for invisibility and muffle it just doubles duration. So it could go either way with rally/courage.

I would guess it increases all affects of the spell

If you play on PC, you can open console, click NPC, type "getAVinfo health" before and after casting courage spell, then write results here.

Thanks, both courage and rally give +35 heal dual cast or not. Wiki seems to be off by 10 points in its description.