gnats around the litter box?

Yeah, it's summer, and no, I don't use the air conditioning, even though it's 85 degrees every day.

And every time I go to clean out Picolo's litter box, there are about 15 little tiny flying bugs swarming around the thing.

I clean the lumps and wet spots out of the litter every day, and add a little fresh litter. I don't dump the whole litter box all at one time, though.
The litter box is the kind with an enclosed "roof".(But no door flap, 'cause it freaked her out )

Is it okay to spray standard bug-killer stuff from the supermarket shelf in the box every day?

I do not think it would be a good idea to spray anything toxic in the litterbox. We have the occasional gnats but we just empty and clean the litterbox with bleach and it seems to keep them under control.