Saz is spot on.
Captainratbag, Saz already posted the quote where a question is raised by the OP about if it is OK to freeze live.
An appropriate diet for an omnivore is not a staple of frozen vegetables. I suggest using a rat and mouse pellet from the pet shop and supplement with any scraps you have. Pellets have been formulated to be a complete and appropriate diet for rodents and will give your mice the bast chance of being healthy and good mums. That being said I don't like many pet shop rodent 'mixes'. They aren't balanced like the pellets, and all that green lucerne chaff gets wasted as the rodents barely eat it. Fancy Feast is a good mix though if you only have a few mice to feed. With only a few mice the rodent pellets will last you AGES though and your rodents will be much healthier than feeding them a random mix of non balanced foods. Frozen peas and corn with a few other scraps and no meat is not in any way a complete or balanced diet.
Inbreeding rodents is the best tool we have to produce consistant, reliable and good natured rodents who are good mothers. When you inbreed you quickly expose any hidden genetic floors, which you then cull, and you are left with babies that do not have floors, who become your future breeders. If you keep inbreeding each generation, your mice will improve as it allows you to select towards all the positive traits that you want in your mice (not eating babies, getting along with other mice, being good mums, being friendly and inquisitive to humans etc).
Being advised that you must use unrelated mice at all times is both unrealistic as well as poor advice. Inbreeding rodents is not the same as inbreeding other species. There are no long term negative effects at all once you have culled off any initial floored mice that pop up.
I euthanise using soda stream Co2 or by donking them on the head. Donking is instant, which is why I like it. Gassing takes a little longer and I only do it when I am culling large numbers. Sarah's method with the BOC cylinder is fantastic because you can regulate the amount co2 being used at each stage. The soda stream is difficult to regulate and there can be times where the rodents don't go unconscious as quickly as you would hope.
All I can say is be careful who you take information from! If this is a school project them you will be interested in learning correct information. You won't find a better source than Saz.
Some people try any idea that pops into their head, and because the rodents didn't die they assume what they do is fantastic and appropriate to pass on to other people as 'the way things should be done'. This doesn't mean their methods are based on fact.