Does the sound volume affects the MP3 file size?
Depends on the details, but typically not. For the sake of discussion, we assume CD quality for lossless, i.e. 16 bit, 44.1 kHz. 


Reducing volume may or may not reduce information. It depends on what number format it's done in. If you convert 16-bit input to, to for example, 32 floating point, no information would be lost. It may still be lost later, depending on what happens to the data down the road
If you do it directly to the 16-bit input data, you have to
basically throw away the LSB (least significant bit), and your
bit depth has been reduced to 15 bit.
This has little or no effect on MP3 encoding. Most encoders start with a target bit rate and then minimize the encoding artifacts within these constraints. If you encode 10s wave file at 320kb/s, the resulting file will be very 3.2 Mbit long. The thing that varies with content is the sound quality and not the file length.
The type of "information" that encoders use are "masking" and "signal to noise ratio". These are largely independent from the overall scale, so shifting volume doesn't affect the encoders much.
Most encoders these days come in two variants: constant bit rate and variable bit rate. With "variable bit rate" you may see the file length fluctuate a bit, but generally not by much. With constant bit rate, the file lengths will always be constant.