Nick I would say that it depends on the size of mice you need... If you the itty bitty ones for hatchlings, then you would want mice... then you could cull all but 2 (moms need to nurse something, I've read it's bad for them if you cull all the pups) and the remaining pups will grow faster...
If you need fuzzies and up, I'd go with rats. Rats are generally friendly(Start out with a friendly pair and you get friendly pups) and have huge litters... I've been switching my snakes up, for three weeks mice one week rats since my rats have been producing like crazy and my ball pythons just don't eat that fast
Mice seem to smell more than rats do, but I change tubs weekly and I use Kiln dried pine which seems to cut down on the smell some. I've also noticed that foods increase or decease the smell... Like Flint River Mills(FRM) seems to increase the smell and the mice seem to drink more. Dog food and Mazuri have about the same level of smell. I'm now using a feed block/seed/left over garden veggie mix now and i haven't noticed a increase or decrease in smell....
I also wouldn't plan on separating the mice when they are pregnant... Mice will often pile all the babies in one pile and take turns feeding the group... So if one mom had 10 pups the others that only had 4 will help her will all those babies... Also moving the mom could stress her out and cause her to reabsorb the litter. If you want to move anything it should be the male, though it isn't necessary.
I haven't noticed any problems with moving the mice to a cleaning container while I switch tubs. I've never had a mouse bite me when messing with the pups, just make sure you touch every single one of them and all the parents so they have your smell. a Mouse bite feels like a pinch, most of the time they don't break the skin but they do some times. I have had a rat bite me when checking her pups and it felt like I took a knife to my finger ... really bad... It did bleed, and it did hurt for the rest of the day and I do have a scar, but I'm a baby when it comes to pain so...