No. It wastes a bit of energy to swing at them, so I usually use a farm implement. Wouldn't mind if left a residue which in large numbers could be machine-ground into jewelry or statues, though. For sale, or keeping on display. I'd like to see that in a future SDV iteration.
You can break the crystals with your sword or scythe which costs no energy. Upon breaking them there is a tiny chance to drop a refined quartz bar, but that is very rare.
I found a slight loss of energy using the sword--but I haven't played in a while, and as my brain sinks slowly into senescence, I could be wrong. Have you checked this out personally? If so, I'll defer to you (on this, at least).
I've noticed no energy loss with a sword, so unless it takes it just for those and I missed it ((which given some of my crystal breaking tirades is possible still but doubtful)) swords do not take energy. Part of why I preferred to find stairs via fighting rather than mining.
Breaking the crystals gives zero experience of any kind. It does, however, have an EXTREMELY low chance (like less than 1% IIRC) of dropping a Refined Quartz. I find them more useful as obstacles to split up mobs so I'm not fighting them all at once, especially if I can split the soot sprites away from the slimes.
Weapons and the scythe definitely costs no energy. I only break the crystals if they are in my way or if there are a lot of them in one place. http://stardewvalleywiki.com/Weapons "Using a tool classified as a weapon does not use Energy."
Breaking crystals is the same as cutting weeds with the scythe or a sword-- literally. I found this out when @KayVeeRiot kindly put some into a chest (using an item spawner) as a gift for me, and saw that the game actually classifies crystal formations as weeds. Literally, they show up in the item database as a variety of weed for purposes of game mechanics. That's why they can be mown down like weeds using a sword. Peculiar, no?
It doesn't surprise me as much as the "weeds" item you get when you dig up a big mushroom tree sapling
They might, if the mine levels were persistent and could be revisited with changes intact. For all we know they actually multiply the same as other weeds, but we never see it because the mine levels always reset between visits. Huh. I wonder if that's worth some experimentation by using an item spawner and putting a few crystal formations up in a safe spot.