Re: i bought the lactic acid…
Dunno how lil_ash prefers to apply but I apply mine using a q-tip. I’ve tried using the fan brush but felt it didn’t give me enough control. Personal preferance. 
As for washing off the peel…once the acid is neutralized, there’s no need for it to be washed off with a cleansar or even a washcloth. From a scientific/chemistry point of view…the acid doesn’t do anything to the skin at all (its like a gun with no bullets). This also goes back to personal preferance. I prefer to take mine off simply with baking soda and water. Lil_ash seems to be happy with taking it off with a gentle cleansar. If you’d prefer using a washcloth, there’s nothing wrong with it (just be gentle!).