Removing and adding skirting boards when installing floors?

Laminate requires an expansion gap. When installed without a gap, laminate buckles up as the flooring heats and expands. Leaving the gap exposed would be unsightly, so the floor trim should be installed over the laminate.

Tile likewise should have an expansion gap when installed over a dissimilar material, like wood subfloor. If the gap around the perimeter of the room is filled with grout, either there won't be sufficient room for expansion and the grout or tile will crack, or the title will contract and you'll be left with an unsightly gap.

Solid or engineered wood flooring can be installed over a subfloor floating (like laminate), nailed, or glued. When floating it requires an expansion gap for the same reasons as laminate. Even when nailed or glued, there's still a little thermal expansion happening. Walls are never exactly straight, and it's difficult to cut the flooring to fit just right without an unsightly gap. It's also difficult to apply a floor finish right up to the edge. So installing trim over the flooring edge is easier and looks better.

In contrast, carpet is stretched between tack strips. There's enough elasticity in the carpet to accommodate thermal changes and cutting inaccuracies (for normal-sized residential rooms, anyhow), so there's no need for an expansion gap, and the edge of the carpet pressed against the wall looks neat. Lacking any technical or aesthetic reason to have a gap, it's less labor intensive to simply leave the skirting in place.

Yes, carpet can be installed without removing the skirting because slightly uneven carpet edges can be stuffed down between the tack strips and the skirting. For wood and tile, straight edges are harder to come by and even a 1/16 of an inch looks bad. The same goes for vinyl, that straight edge will be hard to get. Your alternative to removing the skirting is to add a quarter round piece of molding all the way around the room after laying the vinyl.

Because all the floor coverings you mention expand / contract as the temperature changes, but carpet does not.