Is it safe for 3-week-old puppies to go outside or be away from mom?

Ideally, they should stay with the mother and litter until at least 8 weeks of age. This can prevent behavioral issues as they get older. They can start weaning to solid foods beginning at about 4 weeks of age. If they can stay with the mother and litter while weaning, this would be best for them in the long run.

If the pups are kept warm, there is no problem with her staying away from the puppies for 1/2 hour. If she does not get stressed out by being away from them, you can take her for a walk around the block up to 1/2 hour and later on even longer. It is important that the puppies do not get cold. It is best to take her out after the puppies suckled, and now are sleeping. Make sure they are in a box, so they cannot start crawling around when mom is gone.

No, these puppies cannot be left alone that long. At three weeks of age, they will still be feeding frequently and need their mother with them most of the day. I would see you can have someone stay at your house while you are gone to keep Sophie company to keep her from barking.

Let her have her freedom to decide when she wants to be with her puppies and when not. She needs to go feed them several times a day and probably wants (or should) be with them at night. But during the day some dogs will just go to their puppies to feed them and stay away otherwise. Make sure she has access to them all the time, but do not force her to be with her babies.

Monitoring throughout the 8 week period until the puppies are weaned is recommended, yes, although it may be difficult while you are sleeping. Having the mom and pups in your room and setting an alarm to check on them a few times a night is reasonable. If you are able to get someone to watch them that's great. Otherwise minimize the amount of time they are left alone as much as possible, and try to get a dog sitter if you will be gone more than an hour. Breeding dogs is a lot of work, and it is important to keep track of the mom and pups' progress by checking on them frequently. Here is some more information on birthing puppies https://www.vin.com/veterinarypartner/default.aspx?pid=19239&id=4951546

Yes you can bring him outside for potty training. I would avoid areas that are used by other dogs until a few weeks after his last set of shots. This will give his immune system time to react to the vaccines and develop an immunity to keep him safe form the diseases which other dogs may pass.

Pups should not be away from their mother and siblings until at least 8-10weeks of age to ensure they are fully weaned, have received at least one vaccination and have had an appropriate amount of socialisation with their litter mates and mother, early separation increases anxiety and aggression in many cases. HE should be fed a high quality puppy food and fresh water to drink 15-18months to ensure optimal development and to reduce the risk of growth deformities and joint disease. Be sure to deworm him regularly. The age at which he can be left in an outdoor enclosure depends on your local climate - your local vets office would be best placed to advise on local risk factors for both climate, vaccination schedules and parasite prevention (including ticks) - I recommend you call in and have a chat with them before you bring any puppy home