Best advice: Right now, get him to the vet or the shelter with the note that you may adopt it.

Then deal with the other questions.

Try some Karo syrup in a little puppy milk replacement formula.

"Sleepiness" may be sickness or just hunger & stress.

My own experience with taking a dog to the shelter with a message that I would adopt him if the owner didn't come forward has not been a happy one;
I once found a dog with a hurt foot, couldn't get hold of him because he was terrified, finally called animal control because the dog was running into the street.

They took him to the "shelter", "evaluated" him & found him unadoptable because of the hurt foot & put him on the euthanasia list.

A smart friend recommended I go to the shelter pronto & tell them that he was my dog that had gotten out of the fence.

which I did.

I hope the puppy does well, & I wish you the best.

If the puppy is 5 weeks old you are at an age that he should be fine.

He may or may not have been weaned, so you have options there depending on if he is eating or not. Puppy milk, or goats milk are fine, karo is great to add, chicken broth. At 4-5 weeks they are usually eating puppy mush, which is a mix of soft foods in a bowl. If he's never eaten out of a bowl before the first times will be messy. If he will eat fine out of a bowl you don't have to worry about that. They can eat kibble too at that age.

Potty training will be the same as if you got an 8 week old, they can potty train at this age. Since you work and this is too young to expect a puppy to be able to hold it, you can get a head start if you build a puppy area, small, with a bed area (crate like, box like, with a blanket and soft bed), and then a food and water area, and a potty area, line it all with newspaper and stick a paper towel between two pages of newspaper that had been used to clean up a potty accident, put that that in the potty area, and most puppies as soon as they are of walking age, given a designated potty area, will use that because they do not like to go in their food or sleep areas. I've raised newborns from the shelter for rescue, and had 0 ever go in their bed area, and it was much much easier to potty train them this way.

Mom likely had not started to discipline the puppies. If you have pets, great, that will help a lot with socializing. The issues with a dog this young are going to be behavioral. Teething, mouthing, acting obnoxious. You can mimic a yelp when the pup mouths you so he will learn to not do that. A stuffed animal, or hot water bottle that is warm but not hot, or heck my husband is military too and I've used a canteen with warm water wrapped in a blanket, for the puppy to cuddle with because they like to sleep in piles together.

Everything else is basically the same as having an 8 week old puppy.

Now health issues, worms, but any stray should be dewormed and all puppies normally get dewormed at this age and again at 8 weeks. Parasites like gardia, are common with strays and a vet can tell you if he has that but you would know. If he has no current health issue he can just follow regular vaccinations.