That shread-abox is fine for mice as well.
Other great treats for mice are pretty much any non sweetened cereal (ie no Frosted Flakes or mini-wheats type sweetened cereals) Cheerios are really great treats. They are also much more cheaper than the treats you can buy for mice in pet stores.
In my experience, after the first week of caring for the babies, the adults will be ok with normal short interactions with their 'people'. Depending on how social they were before giving birth will indicate how ok they are with you interacting with them again with babies in the cage.
Offer a cheerio through the bars of the cage or if you have a tank, lower your arm in the cage slowly and as far away from the nest as possible, and offer treats that way. If the mice are really skittish, just talk to them softly so they know you are there, if they come out to see what's up, you can drop a few cheerios close to them (but not too close, don't want them to scare them away) They will quickly associate your presence for food...once they are used to having you around, you can put your arm in and offer food held in fingers or just in the palm of your hand.
After waiting two weeks, I don't see it being any problem cleaning the cage.
How the mice react to having you check out the babies and even handling them really depends on them. Most mice who are ok with their owners around, dont' mind their babies being handled after they are used to taking care of their babies. It is always best not to try and handle the babies while the mother is nursing them or with the babies. They can be very defensive and try and bite, even if it just looks like they are sniffing your fingers, they can still bite if you get too close. But most don't run snarling if they are at one end of the cage and you peak or pick up some babies.
For the most part, nervous mothers only eat their young in the first week after giving birth. By the time the babies are a week old, nervous mothers are typically settled into how to care for the babies. This eating of young I find is much more common amount mouse colonies bred to provide food for reptile and other animals. Feeder mice tend to be much less socialized with humans and thus, stress much easier. Though I have bred mice in the past for feeders, I have kept many as pets as well and I socialize even my feeder mice to some extent (its their babies that are used for feeders and the adults I enjoy as pets). I don't breed mice anymore for feeders as nearly all my snakes are happy with frozen thawed feeder mice.
IN 4 weeks after birth, you will need to separate the male babies from the females. You can keep all the male babies together, just have a large cage with plenty of hiding spots and two or more water bottles and food dishes. This drastically keeps down on fighting amount adult males. However, I have found brothers kept together tend to get along ok but not always. As long as no females are added in with those males, they should be ok...but again, big cage. Spread hides from eachother. This allows the males to have their own 'territories'.
The females can be kept altogether or you can separate some or all of the young females into their own home.

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PHLdyPayne