If putting in a breezeway and redoing the stairs to your current studio isn't in the plans, then I would go with the addition. Wish fire code would be OK with a dumb waiter sort of set up for you. Maybe one of those stair chairs advertised for the older or disabled person would work.
You may be able to use the existing ac and heat source from your home. Or not.
Since I live down here in the desert I would go with a mother-in-law type casit a with solar tubes for additional lighting as it would positively affect the pricing with resale of the property. A shed, wouldn't help much at all.