This job certainly sounds like a great way to interact with a lot of dogs, but there is a lot of education involved in training too. One resource for you might be the
Karen Pryor Academy, which is one of the most highly-respected of the online training schools. Shelter employees get a terrific discount on tuition, and KPA graduates are super-employable as shelter trainers!
The "sticky" thread up top on training resources is also filled with tons of links to articles and videos on all kinds of topics. It's great that you are asking questions, because there are so many resources out there -- great books, great videos, great seminars and workshops, great trainers -- and being eager to learn is the only prerequisite!
Biting (and most dog behavior) has nothing to do with respect. Most biting has to do with fear.
When a dog is afraid or uncomfortable, she can communicate that in many ways. Since dogs don't have words, they can't actually say, "STOP! I FEEL UNCOMFORTABLE!" Instead, they tend to look uncomfortable/fearful. They may move more slowly, seem inhibited, shrink or cower, tuck their tails, flatten their ears, retract the corners of their lips, pant uneasily (or hold their breath), get sweaty paw pads, lick their lips, widen or squint their eyes, or move away from whatever is making them uncomfortable. There are hundreds of signals, large and small, that can tell us when a dog is becoming uncomfortable.
Unfortunately, what a great many dogs experience is that human beings completely ignore all those signs of fear. A great many people do not read dogs very well, because it is a learned skill that takes years to be really good at. And many people tend not to treat fear very seriously even if they do notice it, perhaps because they have been taught to read it as a sign of disrespect, or because they've never stopped to consider that fear is one of the absolute worst emotions a dog can possibly experience. For whatever reasons, many people will continue to do whatever is already making the dog very uncomfortable.
Some dogs may run away, if they have the space and inclination. Some dogs may simply freeze up (which ignorant people sometimes call "submissive," meaning the dog is so overwhelmed with fear that she has basically shut down). Some dogs may try harder to communicate "please, just STOP DOING THAT," perhaps by using distance-seeking signals -- growling, glaring, pushing the corners of their lips forward to signal that they are likelier to bite. Eventually, if their attempts at communication are ignored long enough, some dogs will bite.
All that can take ten minutes or a few seconds, depending on various factors. Learning to identify the moment a dog begins to become uncomfortable is an incredibly useful skill. Because that way, you can start your training long before a dog is pushed to the point of being fearful enough to actual bite.
Dogs bite for other reasons, of course. I have a young dog who gets really mouthy when she's too excited, which has nothing to do with fear and everything to do with hunting/herding instincts (and self-control, which I am happy to say is something she is steadily learning!!). But the kind of biting you are describing is an attempt to escape something the dog finds highly aversive. It does not matter that you don't think it SHOULD be aversive, the dog says it is...and being a good trainer starts with listening to what dogs are telling us!
If respect has anything to do with dog training, it is this: dogs feel calmer, and more able to think & learn, when they are with someone who makes them feel safe. We can make dogs feel safe by listening to them, by behaving predictably, and by making sure that our actions produce consistently positive consequences for the dog. Does that make us a "leader" in the eyes of the dog? Who knows, and who cares...it makes us better friends to dogs, makes our relationships with them immeasurably richer, and makes us more worthy of the kind of trust we often ask from dogs.