
He is not trying to be dominant, dogs are not capable of this. It's a myth. Collies are also very sensitive dogs. It sounds like he is being mishandled and/or misunderstood:
You have said he isn't getting the mental work he needs - that is problem one. This is essential for a dog like a collie, he will be bored and frustrated which will be making things worse.
Problem two is how he is being handled. If you want a dog to lie down, you do NOT do it by taking hold of the collar - that is a recipe for aggression. No matter how gently it is done it is force, it's directed at an extremely vulnerable area of the dog and very close to his face also, it is going to make him feel very uncomfortable and likely threatened. If he wanted the dog to lie down for whatever reason, then a 'down' should be trained in a positive way. If the dog doesn't want to lie down, then he shouldn't be made to and the reason for that needs to be looked at; for example lying down is a vulnerable position to a dog, which could tie in with handling issues; he could be in pain somewhere which could explain the snapping in both situations. If he's worried he could be tense, which could be causing muscle pain somewhere.
The snapping with the other dog is harder to consider without more info but possibly guarding issues over food, or wariness of other dogs which, coupled with a higher emotional situation like handing out treats, is coming out as snapping.
She needs to get a good behaviourist in who can read body language well, understands collies and will NOT advocate any kind of rank reduction/dominance program, but will guide them on the use of positive reinforcement and how to manage the situation and handling methods so that he does not feel the need to snap.
Whereabouts are they? Someone here may know someone good local to them.
Edited to add: all of the above is conjecture based on the brief info given and my experiences/knowledge of dog behaviour generally and collies as well; do not take it as read! They need someone to see the dog in the flesh.