Well said Muggsmagee...lots of good points. I am in no way a "pro' dog trainer but I have had Aussies for the last 25+ years so I can say I have a bit of experience in that category--could not imagine having another breed share my home. Intelligent-yes. High energy-yes. And, yes, there are some Aussies that are aggressive, whether as a result of poor breeding or poor environment, depends.
To have a 5 month old puppy showing that amount of aggression is really a red alert--PLEASE don't get out there and 'socialize' until you and your family get a handle on his behavior at home and with a knowledgeable trainer--the last thing you need is to have him snap at a curious kid who happened to touch him wrong. He truly doesn't think of any of you as the leader of the pack, and since you aren't he's taking that position on his own. I'm afraid that I'd disagree about the being grumpy when woken up giving him an excuse to bite/snap...now if he was woken up by someone kicking him, falling on him, hurting him, that's a different story...I've always gotten my Aussies as puppies, 8-12 weeks old, and was able to work with them from an earlier age, but one of the first things I have always done is to 'mess' with them all the time, while they are eating, putting my hands in their food, giving them bones or toys, then gently but firmly taking them away. I am not saying I am teasing them, holding it out of their reach out of spite or meanness, but I am the boss and this is not a democracy, they do not have a say in it. By doing this when they are younger I have no issues about food possesiveness, nor do I need to take it away, but if I had to I never have to think twice. I can open their mouths (helpful if a pill has to be given that they won't eat willingly), play with their ears, toes, everywhere. They aren't thrilled when the toenails need to be trimmed, but they allow me to, secure in the knowledge that I won't hurt them, and also I am the boss. Don't underestimate his intelligence--he's not going to reason like a human, BUT he can understand more than it sounds like your trainer is allowing him credit for.
I know I will probably not get a fan club from this, but if my dog lunged at someone, especially someone who was smaller than the dog (and it is obvious that is was due to dominance/aggression and not playful, puppy behavior) I would not hesitate to give a very strong, sharp jerk on the choke chain, along with a firm "NO." I would also give the command to 'sit,' refocusing his attention on me, his boss/pack leader--and then reward him with praise and attention when he sat. Again--working with a good trainer, and consistently working at home (with your entire family--who should all be higher up in the pack order than Riley) will do wonders. A crate is a good idea, along with a lot of supervision. I know a lot of folks keep the crates for the entire life of their dogs and that works well for them-- personally I use one the first year then gradually wean them off of it--they are in it usually when we'd leave the house, otherwise they are part of the family, out all the time, (once totally house trained and past the chewing everything stage). I also agree that he should be looking to you for everything-he wants to go outside, you take him out on a leash. Meal time-he sits and waits until you tell him 'ok.' You are the boss. Also, they do tend to bond with one person, but they love their whole family. If everyone participates in their training and care, they are all family. My first Aussie was only mine, when I moved out of state she was my buddy. Smartest dog I ever had, but extremely protective of me, reserved with almost everyone else...until the night she met my future husband (three years before we married)...she hit the floor, on her back, waiting for a belly rub. He fell in love with her right then, and I him...that was in 1987. And he wasn't a dog person.
Your puppy is past the 'cute, get away with everything stage,' and is now in the teen stage, pushing limits and trying to take over top dog position....but it sounds like you are up to the task. Don't give up, but get some help and fast. It's a lot of work...but well worth it in the long run, most things that are worth anything aren't easy, or everyone would have them

Best of luck...from both myself and Macy, one of our two girls...
Oh yeah... NEUTER him ASAP...might help and won't hurt.