Why do the stills don't match the finished product?

The film was probably shot on Tungsten balanced film while the unit photographer probably used daylight balanced film. The still photographer probably used a different exposure as well.

The daylight balanced film makes the tungsten lights appear blue?

No, it makes the blue lights on set appear white. Tungsten film shot under daylight colored lights will appear blue.

Oh ok, so the light they used is white or kinda yellow but the film renders it blue? And they closed the lens so that it appeared darker or how did they do that?

Daylight balanced film is meant to be shot under daylight which has more blue light.
Tungsten filiment lights are much warmer in color and appear orange when shot with daylight film.
Daylight appears blue when shot on Tungsten balanced film.
My best guess is that they used daylight sources with tungsten film to achieve the blue effect. But because most still photographers use daylight balanced film the colors in the stills shot by the still photographer don't look blue. You will notice in your picture all of the orange highlights in Catwoman's costume and the orangeness in their faces. Those are the tungsten lights in the scene. The white highlights are the daylight colored lights. On tungsten balanced film the orange becomes white and the white goes blue.

I'm also guessing that for publicity stills the general idea is to have them brighter for magazines and newspapers where brightly lit photographs are the norm. This was pre-internet so publicity photos had to match the standards of print media.

The cinematographer either selectively underexposed parts of the frame or printed the film darker later.

and what did they do with scenes in the film were the colors appear warmer?

they put orange gels on the lights.

Oh interesting. And when it looks like common daylight? They don't change the film right?