What are the advantages of buying shorter skis?

None.

For a while it was thought that shorter skis made them easier to control, and learn with. But it is pretty clear that it's actually the other way round. With longer skis you have more control (because you have more edge length to use on the snow), and don't have to worry about falling backwards, letting you concentrate on other things (Pizza! Fries! Pizza! Fries!).

Once you're a better skiier, you will also notice the benefits of increased stability, higher speeds, more control at speed, and less effort required to ski.

Always get skis at Eye Height, including for children.

For Off Piste, or Twin Tips, head height, at the very least.