No KOH is soluble. It is an alkali. Just remenber that all compounds of sodium(Na) and potassium(K) are soluble in water without exception.
After reaction the product may be soluble or insoluble. For eg-
1. KOH + HCl ---> KCl + H2O (all products are soluble
2. 2KOH + CaCl2 ---> 2KCl +Ca(OH)2
( Ca(OH)2 is partially soluble)
3. 2KOH + BaCl2 ---> 2KCl +Ba(OH)2
( Ba(OH)2 is insoluble)
No, potassium hydroxide is quite soluble in water and does not precipitate in a double displacement reaction. MOST hydroxides are insoluble, but hydroxides of group 1 metals like potassium are soluble.
I hope that helps. Good luck!