Basically, call the dog by it's name (Rex) not "it"...keep Rex inside until about 6 months of age (depending on his breed)...do NOT leave the puppy outside, unsupervised, at any time *especially at night*...train Rex (examples: sit, stay, come, lay)...potty train him so there are fewer (eventually none) accidents in the house at night....play with and walk Rex daily.....feed him high quality puppy food and make sure he has constant access to water....get some toys for him to play with as well as a leash and collar. Your pup shouldn't have been taken away from his mom/littermates until 8 weeks of age. You will need to make sure you socialize Rex with other dogs, animals, children and adults as soon as possible. Make sure he gets vaccines regularly (dewormer, vaccines, heartworm preventative and rabies shot at 4-6 months). To leave a puppy, this age, outside is asking for trouble (sickness, theft, running away, death, etc.). Please bring the pup inside for a few months. You should have made sure it was o.k. with EVERYONE before bringing a puppy home. Yes, Rex likes it outside. He's a puppy. He wants to explore his new surroundings. It doesn't mean he should be left outside alone all night though.
Bringing him in will probably stop alot of the crying. Placing a t-shirt, or something that is warm and smells like you, will make him feel safe and secure and should also help quiet him down. Make sure he potties before going in for the night. You SHOULD be willing to get up every hour or so and take him out. A pup that young can not hold his bladder/bowels overnight until around 6 months of age.
If, in the end, you can not bring Rex in because of family members then you need to consider giving him to someone that can properly look after a growing, very young puppy.