I don't have any training but I do assessments for shelter dogs on occasion.
I personally approach a dog in its cage first (usually where it is most comfortable) and watch its reaction to my presence. If it slinks off to the back of the cage to avoid me, I just sit down and wait for it to approach me. If it comes right up to me and jumps around, I put a leash on it and take it for a walk, assess the leash manners and any commands it may know. As long as it isn't a massive dog that flails around at the end of the leash, the manners/obedience commands have little to do with whether or not the dog is adoptable.
After it's been walked I take it somewhere quiet and start handling it. I brush it, touch the ears, examine the teeth, pick up the feet and touch in between each toe. If the dog tolerates that, I grab the tail, hold it for a few seconds and drop it; I put both hands on the face and hold it still. I pick it up if it's a small-medium dog. Just basically see how it reacts to all kinds of handling. If it's nervous about any of it, or if it freezes, I don't go any further. I would still consider the dog adoptable if it is uncomfortable with certain things, but like Tessa said, a dog that turns and bites is definitely a "no" until it has been worked with.
I don't really test for food-aggression. I do give the dog a bowl of food and slowly invade its space as it eats. If it continues eating or backs off when I get close, then I just leave it alone. If it freezes or growls as I approach, then I do assume it's posessive of its food. It is SO hard to tell if a dog is food-aggressive in a shelter, so I wouldn't base any concrete decisions on their reaction. I would, however, be more hesitant to place a dog that growls over food with a family with kids. Of course, if the dog has a really severe reaction I wouldn't consider it adoptable unless it gets extensive behavior modification.
Shyness is extremely subjective to the environment. I fostered a GSD that was terrified and distant in the shelter, but confident and Mr. Personality as soon as he got out. I wouldn't consider a severely fear-aggressive dog adoptable, but a shy dog that is distracted/overstimulated or afraid due to the noise is not one that I'd consider a problem. Often times, the softer, more sensitive dogs do have this kind of reaction to shelters. It doesn't mean they'll be shy once they go to new homes.