Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry to hear that Rocky has bloody, mucoid diarrhea and is vomiting blood. I completely understand your concern.
Because he is vomiting, and that means that he cannot take oral medication, he needs to be hospitalized.
He is likely dehydrated because he is vomiting, but if even water is making him vomit you need to take it away from him for now, if he doesn't vomit with licking ice cubes he can have those.
In most cases vomiting is triggered by eating something they should not, too much table food, too many treats or something they find outdoors.
More serious causes of vomiting in puppies include viral (parvo virus for example) or bacterial infections, a dietary allergy or sensitivity, congenital internal organ failure (kidney or liver disease), or a full or partial gastrointestinal obstruction.
In a young dog, especially one that hasn't finished his vaccine series, a viral infection such as parvo virus or a foreign body leading to a partial or full gastrointestinal obstruction would be the most likely cause of the symptoms that you are seeing and both can be deadly. Not all dogs with viral infections run a fever initially, sometimes they are too weak to mount a fever response.
Worms can cause loose stools, but rarely cause vomiting and lethargy.
Because he is young, hasn't been feeling well for more than a day, and he is lethargic ideally he would be hospitalized for intravenous fluids and injectable medications.
I know that you are looking for home care options, but that would be acid reducers like Pepcid, fluids, and a bland diet and if he is vomiting blood be cannot absorb those now. He is just too sick to treat at home.
Most likely this is not contagious to people. The exception would be if this were secondary to a bacterial infection like salmonella. But it is far more likely to be related to a viral infection which is species specific (only infects dogs).
Please let me know if you have any further questions.