The wisps are undissolved silver oxide. If its minor, they may dissolve but its not worth taking the risk. If you're not seeing them when using a stirrer, try increasing the current and see where you get. The temperature you're running your cell at will affect the speed of dissolution as can be expected.
I run at 10ma with a 3.5" long 14 gauge copper wire at 150F and I normally make 120PPM and until the wire gets really thin where its time to change it out, I don't start getting any wisps and what I make is fine. I am NOT using a stirrer.
You want the anode as big as you can afford.
The cathode is used to adjust the voltage across the cell which must be greater than 10V.
I use a length of 24 gauge solid copper wire (insulated) that I've stripped about 1.5" of insulation off the end that goes into the cell. This affords me a fine adjustability where there's still a long enough length of copper in the water that for running at higher temp (me, 150F normally as I'm making higher PPM) even with a little evaporation, the cathode is NOT going to disconnect from the water.
Everyone seems to start with 14 gauge house wire and the issue here is that you barely have to glance the surface of the water to get the cell to the required voltage and thats touchy. After having a little evaporation once and the cathode came out of the water where I had no clue what I made, I thought about it, changed to the thin wire and now I never have less than about 1/2" of submerged cathode, regardless what current I'm running at. If you get THIS much evaporation, something else is wrong over a run.
Others here have followed suit with the thinner wire.
The solubility limit of silver oxide in water at room temp (75F, not colder) is maybe not quite 22PPM. If you're running "cold", you have to make sure that whatever reducer you've added (for higher PPM's you must add it up front), it reduces quickly enough that you never exceed the solubility limit of 22PPM of silver oxide or else you're going to start precipitating out the silver oxide and that wouldn't be good. Never allow the rate of production of silver oxide to exceed the rate of reduction or you will coast over the wire and thats not good.
You could heat the water on the stove or in a microwave to hot enough you can barely hold it (around 150F) and start with that. Over a 2 hour run in a breeze free area, it may not even cool all the way down to room temp.
The other thing you have to be careful about is that say your reducer takes 15 minutes to reduce, after the run is finished, you have to give it 15 minutes plus before the reduction of what you made is finished. I always add a couple extra drops of reducer for good measure once the run is finished. It won't hurt in the least.
Are you using the prescribed 20 drops of 1M sodium carbonate per liter of water?
I've always seen a little black on the cathode but at the end of a run when you wipe the cathode off, its not very much. Copper also won't stay bright - it'll darken a bit with use - thats just surface copper oxide and nothing to worry about.