Tests on sodium peroxoborate
In your first test, the $\ce{KMnO4}$ probably reacted with the $\ce{H2O2}$ excess remaining after your preparation. The ionic equation for the reaction is 

$$\ce{2MnO4- + 3H2O2 -&gt; 2MnO2 + 2H2O +3O2 + 2OH-}$$

The $\ce{MnO2}$ formed is an insoluble black precipitate. Although there is an $\ce{OH-}$, the reaction apparently occurs in neutral and faintly acidic medium as well. 

The result of your second test doesn't seem to be very clear. Sodium perborate apparently reacts very quickly with any water so if you added any water during the second test, it must have reacted to form oxygen gas bubbles (which appear white).