Welcome back and congrats on Tiny. Would love to see pics of him
For the cough, please take him into the vet..it could be any one of a number of things so not sure how to treat until you know. Kennel cough, bronchitis, anything..but it could turn into pneumonia easily if not treated.
Deeply sorry for the loss of your precious Jet
I'm glad you found Tiny to give you some puppy lovin to help you heal!