For bulk material it is almost constant. When u make a solution u have diluted the material in a way that new bonds have been formed between the related atoms and the solvent. This can change the nature of the material. Even in solid state the bandgap is a function of sevaral factors like growth conditions and post-growth ones. Thin film naturally follows the same way and some factors like the degree of amorphousness, thickness, strain/stress etc. can change the bandgap let alone the liquid phase. 

Mostly not... but even if it changes... it doesn't change that much depending upon your synthesis protocol and particle size and morphology etc.

Good luck!