Any ideas on how to install a bed skirt and keep it straight?

https://youtu.be/jUcQVC_qrXk

They do sell pins/nails that hold a bed skirt onto mattresses. Find at fabric stores.

There is no easy way, mattresses put up a good fight! Been there, done that. Enlist a group of 2 to 3 people and work together.

I do think you'll have to remove the mattress to successfully get the bed skirt on, but this Hometalk trick makes it pretty simple-https://www.hometalk.com/diy/decorate/rooms/diy-no-sew-drop-cloth-bed-skirt-17550386

You can buy bedskirt that are "wraparound" so you don't have to move the mattress. May need to pin up in a couple of places.

Consider using a length of fabric and attaching it to the bed frame instead of under the mattress.