Jill, I am quite sure Kian would say to NOT carry the puppy around (unless you meant in his crate, i.e., moving the crate from room to room as you go)!!! I also don't think sneaking off is a good idea as then he will never learn to trust you.
Throwing up from crying is a bit unusual in my experience, but he is very young. I would take him to the Vet as soon as possible, and I would also make sure you are feeding him the proper food. What are you feeding him right now? In our Health Section, there are many discussions about nutrition and the various stages of feeding a puppy needs. For example, when Oliver came home, his breakfast was baby pablum (rice formula) made with GOAT'S milk....Cow's milk is very hard on a puppy's tummy. He was nearly 9 weeks old! He also had a lunch, and a dinner, and two snacks per day. His dinner included 1/4 of a teaspoon of Cod Liver Oil, and bit of cooked meat. He was on this diet until 16 weeks, it then changed until he was 6 months old, which then changed again once he attained the age of 1.
Ringo (cute name and for a cute reason) needs to be thoroughly vetted for parasites, and if your daughter can't provide proof., i.e., a signed document indicating the vaccines he supposedly had, then Ringo will probably need to be vaccinated. I can't remember where you live, or if you even said. I know that there are certain parasites that live in the grass especially in the summer, that can make your puppy quite sick.
Kudos to you for waiting until he was settled to let him out of the crate!!! That was perfect. This may take a few weeks, so be prepared to not being able to do everything you wish to do. I think I would leave the dishes for your daughter!!!!
Perhaps she can pick up the slack since you are busier than ever, as well as pay for his Vet visit!!!
I'm sure Kian will be along to offer some sound advice.