Sure, just make sure you have a safe place to put them temporarily while you clean the cage. Just something they can't get out of, inaccessible to other pets (predators). I just use those plastic boxes/cages from pet stores with paper towel on the bottom...
yes, dont worry, i mean they probably dont wanna be picked up but yeh if you need to move them as long as your picking them up normally thats fine. you know sometimes you have to do whats necessary, like checking theyre tummies and stuff, you know if they could talk they'd probably say no but we have to sometimes.
You dont have to clean around them just move them. And they'll go somehwere else.
Yeah I usually clean the cages during the day. (when they are asleep) I just throw them into a container either a ziplock container or a cricket keeper. Then put them back into a dark place so they can go back to sleep.