Feral cat with new kittens: Should I bring them inside for safety?

I would leave her alone as much as possible. It will already be stressful for her to be inside, and visiting Cali a lot will make her more stressed. I would give her a nesting box to lay in, and only go in once a day to fill her food bowl and water bowl.

That is very kind of you to want to help this cat and her kittens! Yes, it is fine to touch them, but I would be careful around Momma. Some female cats can be very possessive of their kittens, and she may act aggressive if you try to move them. I would recommend giving the Humane Society a call, they will have the tools to trap her and her kittens safely.

Possibly - especially if she is now pretty tame. If she is easily spooked or still not particularly attached to you it might not be a good idea. I recommend starting with short supervised trips outside. You should only feed your cat indoors and offer food during specific meal times to make sure you get her to come back in at night. You can also practice calling to her and giving a treat. Keep in mind that living indoors is much safer for cats and there are risks in going outdoors: infectious diseases, fights with other cats, and cars and dogs are a few of the threats. If she has not been vaccinated for the Feline Leukemia virus I would recommend completing that vaccine series before letting her outdoors where she could contact other cats.

You can move the kittens - if you are planning to socialize the kittens and either keep them yourself or adopt them out it would actually be ideal to handle them when they are young. It is unlikely that the mother will reject them, but she may try to move them back unless you cut off access to the other room. Depending on how social the mother is you may need to wean the kittens early to be able to fully socialize them to people - ask your vet for pointers. http://4fi8v2446i0sw2rpq2a3fg51-wpengine.netdna-ssl.com/wp-content/uploads/2014/10/Socializing-Kittens_update.pdf

I would set up a warm bed for them, and put in something like a hot water bottle covered in towels. They are just over four weeks old, and while it's time for them to wean, they are probably better with their mother. If the temperatures keep dropping, however, I would move them indoors.

She feels that her kittens may be safer in the house. She might want to move them from thee cat house. Can you bring the cat house close to the house or inside or on your porch (if you have one?)?. She will also now need to eat kitten food, which is more nutritious than her normal food.

Mom thinks this is a safe place, and they will be fine there. Try to make sure she has access to fresh food and water and they'll be fine. Take them in for deworming and vaccinations at 6-8 weeks.