Is there any problem soldering a device while it's powered on?
Typically, no - if you're careful. You can use gas-powered iron or just unplug the usual one from its power source just prior to making a joint (that's what I do). It is actually quite convenient during debugging since you just resolder homemade probes from one test point to another instead of figuring out how to re-clip fancy expensive ones without shorting adjacent pins. For parts with 0.5 mm pitch and less soldering is the easiest way to attach a probe since you'd have to probe under a microscope anyway and while you move your eyes from the board to the display of your instrument you can also easily move a probe if you're holding it with your hand.

"Careful" is very important. No matter how good you think you are never solder live line-powered circuits, under any circumstances. At my job I don't have an iron in a room where I work with SMPS stuff.