How do I remove and reinstall baseboards (Skirting Board) without damaging them?
Pretty easy DIY job. All you'll need is: a hammer, old metal putty knife, utility knife, small pry bar, pliers, and a pencil. For reinstallation, a finish nail gun really makes the job go fast, and a nail set is also useful.

The typical procedure:


Cut away any caulk on the top edge and corners.
Check any inside corners to see if they are coped rather than mitered. If they are coped, you have to remove the coped side first.
Use an old putty knife to pry back slightly between the top edge and the wall. Focus any prying near the studs or where ever you find nails, pulling where there isn't any nail too far will risk cracking the trim.
In the crack, use a small pry bar so that all the force is applied to the back of the trim, and not the top edge. Work you way down, one end to the other. To protect the wall, pry against the putty knife rather than directly against the wall itself.
Pull any nails in the trim through the back with a pair of large pliers rather than hammering them back out the face where you'd damage the wood and paint.
Number the back side of the trim and the bottom of the wall where the numbers will be covered up upon reinstallation.
For reinstalling, a nail gun works best with finish nails. One nail per stud and two at the ends is usually enough for basic trim. Use a nail set to sink any nails slightly below the face of the trim. If a nail starts to bend, pull it with the pliers, using the putty knife to protect the trim. Do not bend the nail over and hammer it in sideways.
Paint any trim before it goes on the wall, use a small bit of spackle or wood putty on the nail holes, caulk the top edge and joints, and all that's left is a bit of touch-up paint.


Pitfalls:


Prying too hard and too far away from a nail, causing the trim to crack.
Applying force to the delicate top edge.
Trying to pull the flat end out of a coped corner first, breaking off the delicate coping.
Not cutting away any caulking, resulting in the paint and drywall pulling away with the trim.
Trying to knock out any finish nails back through the face of the trim, causing lots of damage to the trim.
Nailing in a groove or somewhere that you can't get a smooth spackle finish.
Nailing so close to the edge that the trim splinters.