It depends on the hamster. Our first syrian would happily go to sleep in his pet carrier with a blanket over the top when I wanted to sort his cage. Our last Syrian wouldn't and was desperate to get out. With him I set up the playpen and left him in there so I could easily pop in and check on him every few minutes. You need a secure playpen for that. Either way I find it better that they are in a different room from the cage as they know you're messing with their cage and it upsets them!
To get her out, try a tube with a bit of cheese or cucumber at the far end and put the other end over her house door. When she walks in the tube, put your hands over each end and lift the tube out. Have something next to the cage - eg hamster ball - and take one hand off and put that end in the hamster ball so she walks into the ball - then pop the lid on. You can then transport her where you want to put her while you sort the cage. But I wouldn't leave a hamster in a ball while doing it.
Also don't do a full cage clean
Just the substrate. You can do any other items another time - it helps that something always smells familiar or they get quite stressed. I also keep back a bit of the old substrate that's clean and spread it on top of the new - it helps them settle a lot easier after the change.