Cutting and resoldering CPU chassis fan power cables
Your plan looks fine, although it is a good idea to cut, then strip. Cut staggered (see below). 

Some suggestions:


Keep the exposed area as short as possible. 5-15mm (1/4 - 1/2 in) off each wire depending on how big your soldering iron tip is (and how confident you are!)
Tin both ends before join
Thread heatshrink on wires before join (if you're using heatshrink)
Clean iron often on wet sponge to get dry solder off the iron.
Feed solder onto joint while you heat with iron. 
Stagger the soldered joints, for example, + is where the soldered joint is:

wire 1: -------+----------------------

wire 2: --------------+---------------

wire 3: ---------------------+--------


This will stop the joints from shorting on eachother if your insulation comes loose. It will also reduce the insulation thickness, since the thicker sections are staggered.

Insulation options: 


Heatshrink:
Three small gauge heatshrink pieces for each individual wire, one larger gauge over all of them together. This is really a nice and neat way of insulating.
Insulation tape:
Small pieces of insulation tape to insulate them separately, then 1 bigger piece to insulate together over the top. You don't need a lot of tape, a little goes a long way. 


Nice tutorial (with pictures):
http://www.personal.psu.edu/djn5065/wires.html