Nope. You can on the other hand get nobility. This happens about 0% of the time though As a hansel I would unload my weak troops on the easiest quests and get around 1nobility per two unloads. So basically no. There is no way to get crystals or nobility without paying, being very lucky or certain occasions (e.g. Christmas, halloween or servers go down)
Yes. Doing quests sometimes drops an xstal, but it is VERY rare. Nobs also drop, and less rare, but still very rare. Sometimes the devs give us some xstals, for example at Christmas.
No just a very very very small chance on getting one when u do a quest after completion and I quote very small like this small