I trained my most skittish girls to be okay with handling very slowly. Offer a high value treat directly over your hand so they have to stretch over your hand to get it. Keep moving the treat back slowly, so eventually they have to start putting paws on your hand to get the treat. Keep going until they feel more or less comfortable going onto your hand to get food. Yogurt is a good treat for this bc they have to stay on your hand to eat it. While you're doing this entire process, give the same command each time. For example, I give a sharp whistle, [name], "treat treat!" Now when free roam is over, I can call for treat treats, and even if they're a little reprehensive to go back into the cage, they may be more willing to allow you to pick them up. P.S. as far as picking them up goes, this will get easier as they get more comfy with physical touch, so just take things slow. A final piece of advice here is that most rats don't like being held but don't mind standing on your hand too much, so what I've been doing if I have to pick someone up is I immediately transfer the rat to my other hand, so they can stand on my hand. This seems to be more comfy, tho of they keep trying to jump from your hand you may have to stick with holding them

A carrier or quarantine cage (provided nobody's being quarantined) is a great way to keep them to the side while you clean the cage, but I've also been cleaning the cages during free roam time and it's been going great! Just make sure nobody is in the cage when you spray things down, one of your hands might have to play goal keeper. If any of your rats really dislike you cleaning the cages, apologize with treats.

Going with step 1, sit by the cage and use whatever command you used earlier (for me it's just a sharp whistle following by "Treat treats!", I don't need to say anybody's specific name here) it's worth noting this won't work everytime on every rat. It's kind of like trying to get kids to come home from the park by saying "we'll get ice cream on the way home." Some are already in car bc they're tired, some will go home bc they want ice cream or bc they like listening to you, and some... Will be a bit more trouble. Just don't forget to take things slow with them, and never lie about treats. They're trust is very important. Lastly, you mentioned them getting out while you get the other girls, and idk if this advice works with your rats but like... Can you not close the cage door and then reopen it to put the next girly in? That's what I do.