Hello. I am sorry to hear your new little one is not feeling well. My name is ***** ***** I would be happy to help.
At this age even though she has had some vaccines she is not fully protected against things like parvo and distemper as it take 3 rounds of vaccines for full protection.
Loose stool in puppies this age is most often due to parasites meaning worms or protozoa such as coccidia or giardia.
The runny nose and cough can indicate an upper respiratory infection.
When ever a pup does not eat that is a concern because at this age all they should be doing is playing eating and sleeping. When a small bred pup does not eat every few hours they can not maintain the proper blood glucose levels and the glucose can drop dangerously low. They can also become dehydrated very quickly from diarrhea at this age.
That temp is at the low end of the normal range so that is not concerning. Normal for a dog is 100.5-102.5
Should you be concerned yes Unfortunately it sounds like little June Bug has not had a very good start and may not have come from the best environment. A good breeder would never let a pup go that had loos stool or any respiratory issues.
I would suggest getting her in to see your vet as soon as they open tomorrow. For tonight try to offer some boiled chicken or hamburger to get her to eat. If she will not eat or become lethargic, has any muscle twitching, shaking, or declines in any way then I would have her seen by an ER vet tonight.
I am sure with the vets help and your loving care you will get her on the right track.
Please reply back if you have additional questions and I will be happy to continue. If you have no additional questions please take a few moments to provide a rating so that I am compensated by justanswer for assisting you today.