Is it possible to lower the sound of the Robot while moving?
I know how to do it for most things in the game, but not the robot character.

It is the game effect sound category.

I've found only two sounds in the prototype: "eat" and "heartbeat". Not sure where the running sound is set, I'd assume this would sound the same for all characters -- is that true?
Meanwhile, why don't you get yourself some Safety ear muffs? I just had a quick look at their settings, and there are startup settings "Mute Prototypes by name", and "Volume override". The prototype name of the new character is "IRobot_character_skin", so you could give that a try. However, I'm not sure if that would really work. If it does, feedback would be welcome, so I could put that on the info page and add a dependency on eradicator's mod.

Turning the walking sound down in settings works. Safety ear muffs doesn't. It sounds like they are speeding up the normal walking sound quite a bit from the normal character.
Here is a link to a discussion of a few years ago about the issue. https://forums.factorio.com/viewtopic.php?t=33358
Looks like it is programmed in the code and not meant to be changed.

Too bad, but thanks for testing this anyway! :-)

No problem. It looks like turning the sound of the Walking Sound in settings works ok, but it has to be done manually when changing back and forth between characters. Oh well.