First, some very general information, called The Rule of Three
What should I expect?
You can expect that it will take your dog some time getting used to the new routines and adapt to his new environment. The ‘Rule of Three’ means that you can gauge the time it
might take for your dog to fully acclimate to his home in threes: three days, three weeks, and three months.
Remember, this is just a general guideline.
At 3 days…
The first 3 days are the initial adjustment period as your dog transitions from their previous home to your home. Your home is new and exciting, with different activity and spaces than they are used to. It can be overwhelming for many dogs.
Your new dog may sleep a lot in those first few days or he may be so amped up
on excitement that he is easily aroused and difficult to settle down. He will want to check out all the new smells and investigate his new digs. He won’t know what you expect from him, where to go potty, or whether he’s allowed on the furniture; he won’t know that your shoe is not actually a chew toy, or that the kitchen trash is not where he is supposed to find his dinner.
These first few days require an immense amount of patience on your part. Take a deep breath and remember that your home is like Disneyland to your new puppy. He will settle in to your routine if you give him time and patience. It won’t happen overnight, and if they're available, attend positive-reinforcement training classes to help him learn better manners, but take comfort in knowing that it gets better!
At 3 weeks…
After 3 weeks, your dog is probably getting used to your comings and goings, learning the daily routine, and starting to figure out when the next meal is coming. He’ll learn that you walk at the same time every morning, and that he gets to go out for regular potty breaks. You’ll start to see more of his true personality and less of his initial response – whether that was fear, excitement, stress or a combination of all three. You will become more familiar with his true personality as it blossoms in his new environment. It won’t be completely smooth sailing, but the bumps in the road will be less frequent and less stressful.
At 3 months…
At 3 months, most dogs know they are “home.” It’s a process to get there, but with patience and a sense of humor, the two of you can scale the mountain together and enjoy the journey toward a great relationship.
Love Has No Age Limit, Drs. Karen London and Patricia McConnell
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That Teddy is crying in the night or when you leave is completely normal at his age. He is just a baby, and as fjm said, his needs are just like a human baby's.
He will cry when scared, when lonely, when hungry, when he needs to eliminate, It's up to you to determine which is causing his distress and then take care of it for him. If he's scared or lonely go to him so he knows you're there. As fjm said, iIf you can manage it, it's so much better to have him in the same room with you at night. You'll know if needs to go out, or if he's ill, or just scared and lonely. This phase of intense needs will diminish as he gets older, and with puppies, that happens much faster than humans.
Regarding his needs for elimination, his internal systems are very small and immature so they can't really "hold" their eliminations reliably til they're around 6 months old.
Until then the general rule is that he'll need to go out at least every 2 hours at 2 months old. He'll also need to go out after every meal, after every drink, after every nap, after every play session, pretty much after anything.
At 3 months, the interval may go up to 3 hours as his internal systems start maturing. The interval can increase by about 1 hour per month of age. This is not a schedule to follow with timers and such, just a guideline.
Until his body matures it's not unusual at all for him to need to eliminate 2-3 times during the night. The good news is that he will grow out of that quickly, comparatively speaking. It will be hard on you, sleep deprivation is a part of puppyhood for most of us

.
As for crying when he's left alone, that's also normal. Do you have any way of knowing if he settles down after a while or if he cries continuously til your return? Fjm's suggestion of very brief absences, just stepping out of the room, or out the front door for only a short time will help him learn that it's normal for you to go, for him to stay, and most importantly, that you will come back! Having the Kong toy with part of his daily food in it can give him something to occupy his attention for a bit. Make sure that no choking hazards are available when no one is there to watch him.
What size is Teddy? Is he a toy or miniature or standard? Each size has different growth rates so this is helpful information for us to know, to help you and Teddy.
Puppies do best on free access to water, which
may be withheld only about an hour or so before bedtime if he needs to eliminate very soon after bedtime. They do best on 3 meals a day to help regulate their internal systems and keep nutrients coming at a regular pace til they're at least 6 months old.
If he's crying because he's lonely or scared at night and doesn't necessarily have to eliminate, and if you're not able to keep him in the same room where he can hear and smell you, even have you just reach down to give a gentle stroke, there is something which might help. Some members have used this very successfully.
It's called
SmartPetLove Snuggle Puppy Behavioral Aid Toy
and is a wonderful tool to help soothe a puppy. I don't know where you all are or what access you have to the retailers I'm used to in the US but I'll put the Amazon link here so you can see the item description.
I don't know why the Amazon links sometimes say "Robot Check" but it is safe to click on. If you'd rather just search online yourself just use the name over the picture.
I would say to reassure him rather than let him cry it out on his own. You want him to learn to trust you to look after him. That's a large part of what the bond between you all will be built from.
Please stay in touch!