One caution...look into the run of existing pipe and see how it's supported. When you snap the CI, you could possiby have the sections of pipe shift or even fall. Shouldn't happen if it's properly supported...but then, you never know.
A judiciously placed quick clamp or two can help.
Soil pipe cutters can usually be rented at a rent-all.
For the order of work, do all the peeking inside the walls and demo before you snap the pipe. That'll help you figure the real path the repair will take. Get all the fittings, then snap, then reroute and repair.
Do you know why the pipe was run outside the wall to begin with? Is there a beam above or below that prevented an enclosed run?