I know supports can be used to hold stuff up... but can they keep something from falling?
By 'support', are you referring to the building called a 'support', or to actual constructions made from walls/floors/whatever? Both Supports and constructions will hold up walls and other such structures both above and below them, so you can use a support to hang a structure from an overhanging roof. Constructions (walls, floors, etc) will also hold up structures on adjacent tiles on the same level, but I'm pretty sure supports will not.
For example, if I build a 3x3 wall block in midair held up by a support, I know that will work. But if I were to create an overhang, say, 10z levels high, and build a 3x3 wall block underneath the overhand, could I attach it to a support, remove scaffolding, and then it would stay in place until I destroyed the support?
Yes, this will work. Supports hold up objects below them if the top is anchored to something.
Also if I drop a column into a lake, will it penetrate the floor or will it just rest on the bottom? And if its top is higher than the water level and is otherwise completely closed off will the water not enter? And what happens to the displaced water?
The falling column will punch through the bottom of the lake only if there is an open space directly beneath the lake floor. Otherwise, it will not punch through.
If the column is made from constructed walls, it will deconstruct into its component materials when it hits the floor of the lake, and you'll have a pile of stones or blocks or whatever on the floor of the lake. If it's made from naturally occurring stone or cast obsidian, it will become a column resting on the bottom of the lake.
The water that used to occupy that space will teleport upwards, appearing directly above the top of the column. If the falling structure is a tube whose top is above the level of the lake, water won't be able to flow from the lake into it, but the water that occupied the space that the tube fell around will still be there, and some of the water that was displaced upwards may fall into the tube after it falls. You'll have to pump or drain that out before you can use the tube as an entrance.